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METHODS   OF 
MEASURING   ELECTRICAL   RESISTANCE 


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Electrical  World         TheEn^nGeringandMiiung  Journal 
Engineering  Record  Engineering  News 

Railway  Age  Gazette  American  Machinist 

Signal  Engineer  American En$rm 

Electric  Railway  Journal  Coal  Age 

Metallurgical  and  Chemical  Engineering  Power 


METHODS  OF  MEASURING 

ELECTRICAL  RESISTANCE 


BY 


EDWIN   F.   NORTHRUP,   PH.D. 

Mem.  A.  I.  E.  E.,  Am.  Electrochem.  Soc.,  Inventors  Guild, 

Am.  Phys.  Soc.,  Franklin  Institute,  Fellow  A.  A.  A.  S. 

Palmer  Physical  Laboratory,  Princeton  University 


McGRAW-HILL   BOOK   COMPANY 

239  WEST  39TH  STREET,  NEW  YORK 
6  BOUVERIE  STREET,  LONDON,  E.G. 

1912 


Engineering 
Library 


COPYRIGHT,  1912, 

BY    THE 

McGRAW-HILL  BOOK  COMPANY 


Stanbopc  ipress 

F.    H.GILSON   COMPANY 
BOSTON,  U.S.A. 


PREFACE. 


THIS  treatise  contains  a  compilation  of  many  methods  of 
measuring  electrical  resistance,  most  of  which  are  fully  described. 
Some  of  the  methods  are  new  and  are  described  here  for  the  first 
time.  Several  are  illustrated  with  records  of  sample  measure- 
ments. While  it  is  not  claimed  that  the  work  is  exhaustive,  the 
author  has  selected  for  presentation  all  methods  which  in  his 
judgment  are  useful,  for  commercial  tests  and  measurements,  for 
purposes  of  instruction  in  educational  institutions  and  for  appli- 
cation in  technical  and  research  laboratories.  Rules  for  the 
estimation  of  errors  are  briefly  considered  in  the  first  chapter. 
One  chapter  is  devoted  to  methods  of  measuring  temperature  by 
means  of  resistance  measuring  apparatus,  and  in  another  chapter 
methods  are  considered  for  locating  faults  upon  telephone  and 
other  land  lines.  While  few  descriptions  of  specific  types  of  in- 
struments are  given,  two  chapters  are  devoted  to  a  consideration 
of  the  broad  principles  which  should  apply  when  designing,  select- 
ing and  using  apparatus  intended  for  .the  measurement  of  elec- 
trical resistance.  An  appendix  contains  data  and  information 
useful  in  connection  with  the  subjects  treated.  Methods  employed 
for  the  absolute  determination  of  the  ohm  are  not  considered 
because  few  persons  have  occasion  to  make  this  determination. 

In  the  examples  recorded  to  illustrate  specific  methods,  it  may 
at  times  appear  to  some  that  the  precision  obtained  is  unsatis- 
factory. The  measurements  recorded,  however,  are  real  and  not 
hypothetical  cases,  and  they  were  made  under  such  working  con- 
ditions as  ordinarily  obtain.  They  are  thought,  therefore,  to  be 
more  instructive  than  specially  selected  cases  where  the  measure- 
ments have  been  made  with  unusual  skill  and  care  resulting  in 
exceptionally  high  precision. 

The  author  has  felt  justified  in  writing  upon  methods  of 
measuring  electrical  resistance,  because  for  over  twenty  years  he 
has  been  engaged  in  electrical  measurement,  and  for  over  seven 
years  he  was  connected  with  The  Leeds  and  Northrup  Company 


257915 


VI  PREFACE 

of  Philadelphia,  Pa.  (but  with  whom  he  now  has  no  association), 
which  manufactures  electrical  resistance  measuring  apparatus. 
He  hopes  by  recording  the  experience  acquired  he  may  benefit 
those  who  are  interested  in  similar  lines. 

Doubtless  this  book  is  not  free  from  errors  and  defects.  Any 
reader  noting  such  will  confer  a  favor  upon  the  author  by  pointing 
them  out. 

The  author  acknowledges  his  indebtedness  to  Mr.  J.  W.  Wright 
of  the  Bell  Telephone  Company  of  Pennsylvania,  for  his  careful 
reading  of  the  chapter,  "  Elementary  Principles  of  Fault  Location," 
and  for  the  valuable  suggestions  which  he  offered  for  its  improve- 
ment. He  adds  his  acknowledgment  to  The  Leeds  and  Northrup 
Company  for  the  loan  of  electrotypes.  He  further  takes  this 
opportunity  to  express  gratitude  to  his  wife,  Margaret  Stewart 
Northrup,  for  her  encouragement  to  proceed  with  the  work  and 
for  her  unremitting  assistance  in  the  preparation  of  the  manu- 
script. 

EDWIN  F.  NORTHRUP. 
PALMER  PHYSICAL  LABORATORY, 
PRINCETON  UNIVERSITY. 
December,  1912. 


CONTENTS. 

PREFACE. 

CHAPTER  I.  —  EXTENT,  CHARACTER  AND  PRECISION  OF 

ELECTRICAL  MEASUREMENT.     THEORY  OF  ERRORS. 

OHMIC  RESISTANCE. 

ART.  PAGE 

100.  Electrical  Measurement 1 

101.  Some  General  Principles 2 

102.  Comments  on  Accuracy  and  Method 4 

103.  Elements  of  the  Theory  of  Errors 7 

104.  Application  of  the  Theory  of  Errors 12 

105.  Comments  on  Ohmic  Resistance 16 

CHAPTER  II.  —  RESISTANCE  MEASURED  WITH  DEFLECTION 
INSTRUMENTS;    VOLTMETER  AND  AMMETER  METHODS. 

200.  Assumptions 20 

201.  Voltmeter  Method.     Circuit     Includes     a     Known     Resistance. 

Method  1 20 

202.  Voltmeter  Method.     Circuit   Includes  an  Unknown  Resistance.      22 

203.  Voltmeter    Method.     Circuit    Includes    a    Known    Resistance. 

Method  II 23 

204.  Comparing    Potential    Drops    with    a    Deflection    Instrument; 

Special  Case 24 

205.  Voltmeter  Method  Using  a  Shunt 26 

206.  Deflection  Method.     Resistance  Measured  by  Substitution 27 

207.  Voltmeter  Method.     Circuit  Forms  Loop  of  Three  Unknown  Re- 

sistances, Two  of  which  are  to  be  Determined 30 

208.  Limitations  of  Voltmeter  Methods 34 

209.  Resistance  Measured  with  a  Voltmeter  and  an  Ammeter 35 

210.  Remarks  Upon  the  Methods  of  Chapter  II 37 

211.  Ohmmeters  and  Meggers 38 

CHAPTER  III.  —  NULL  METHODS.     RESISTANCE  MEASURED 
BY  DIFFERENTIAL  INSTRUMENTS. 

300.  Remarks  on  Null  Methods 40 

301.  Properties  of  Differential  Circuits 41 

302.  Illustration  of  the  Practical  Advantages  of  Differential  Circuits .  . .  45 

303.  Differential  Galvanometer  Used  with  Shunts 48 

304.  The  Differential  Telephone 50 

vii 


Vlii  CONTENTS 

CHAPTER  IV.  —  THE  WHEATSTONE-BRIDGE  NETWORK. 
SLIDE-WIRE-BRIDGE  METHODS. 

ART.  PAGE 

400.  Network  of  the  Wheatstone  Bridge 51 

401.  Uses  of  the  Slide-wire  Bridge 54 

402.  Comparison  of  Resistances  by  Modified  Slide- wire  Bridge 58 

403.  The  Carey-Foster  Method 61 

404.  Galvanometer  Resistance.     Measured,  Using  the  "  Second  Prop- 

erty" of  the  Bridge 69 

405.  Calibration  of  Bridge  Wire 70 

406.  The  " Kelvin- Varley  Slides" 75 

CHAPTER  V.  —  WHEATSTONE-BRIDGE  METHODS.     VARIABLE 

RHEOSTAT.     ARRANGEMENTS  OF  RESISTANCES. 

PER  CENT  BRIDGE.     SUGGESTIONS 

FOR  USING  BRIDGE. 

500.  Wheatstone-bridge  Methods  with  Variable  Rheostat 78 

501.  Arrangements  of  Resistances  in  Wheatstone-bridge  Rheostats.  .  .  79 

502.  Rheostat  Coils;  Classical  Arrangements 81 

503.  Northrup's  Four-coil  Arrangement 82 

504.  Five-coil  Combinations 86 

505.  Decade  System  of  Feussner 87 

506.  Decade  System  of  Irving  Smith 87 

507.  Multiple  Arrangements 87 

508.  Arrangements  of  Resistances  for  the  Ratio  Arms  of  Wheatstone 

Bridges 89 

509.  Schone's  Arrangement  of  Ratio  Arms 91 

510.  Nonreversible  Ratio  Arms  Adjustable  without  Contact  Resistances.  92 

511.  Wheatstone  Bridge  Arranged  for  Reading  in  Per  Cent 93 

512.  Remarks  upon  the  Use  of  the  Wheatstone  Bridge 94 

CHAPTER  VI.  —  THE  MEASUREMENT  OF  Low  RESISTANCE. 

600.  Introductory  Statement 100 

601.  Low  Resistance  Measured   with  an  Ammeter  and  a  Millivolt- 

meter  101 

602.  To  Measure  the  Resistance  of  Sections  of  a  Closed  Circuit;  General 

Method 102 

603.  To  Measure  the  Resistance  Between  Two  Points  on  a  Bus-bar     .  .  106 

604.  Measurement  of  the  Current  in  a  Bus-bar 108 

605.  Measurement  of  the  Resistance  of  Underground  Mains 110 

606.  Comparison    of    Low    Resistances    by    the    Modified    Slide-wire 

Bridge 114 

607.  Comparison    of    Low    Resistances    by    the    Carey-Foster-bridge 

Method 114 

608.  Comparison  of  Low  Resistances  with  a  Potentiometer 115 

609.  The  Kelvin  Double  Bridge;  A  Network  of  Nine  Conductors 115 

610.  Theory  of  the  Kelvin  Double  Bridge 117 


CONTENTS  ix 

ART.  PAGE 

611.  Sensibility  Which   Can  be  Obtained   With  the   Kelvin   Double 

Bridge , 120 

612.  Methods  of  Applying  the"Kelvin  Double  Bridge  Principle 123 

613.  Plan  of  Procedure  for  Making  and  Recording  a  Measurement. . . .  126 

614.  Sample  of  a  Low-resistance  Measurement;    Resistivity  of  Mag- 

nesium    127 

CHAPTER  VII.  —  THE  DETERMINATION  OF  ELECTRICAL 
CONDUCTIVITY. 

700.  Standards  of  Conductivity;    Their  Relation.     Useful  Formulae.  .  132 

701.  The  Measurement  of  Conductivity 140 

702.  The  Hoopes'  Bridge  for  Conductivity  Determinations;    Described  140 

703.  The  Hoopes'  Bridge;   Operations  Required  for  Using 143 

704.  Precautions  to  Observe  in  Using  Hoopes'  Bridge 144 

705.  Other  Methods  of  Measuring  Conductivity 145 

706.  Equipment  for  Conductivity  Determination;    The  Standard  Re- 

sistance Variable 145 

707.  Method  of  Using  Variable  Resistance  Standard  for  Conductivity 

Determinations 147 

708.  Method  of  Calculating  Conductivity  from  Resistance  Data 148 

709.  Conductivity   Determinations   with    Fixed   Resistance   Standard 

and  Variable  Ratios 150 

CHAPTER  VIII.  —  THE  MEASUREMENT  OF  HIGH  RESISTANCE. 

800.  High  Resistance  Specified  and  Described 152 

801.  Wheatstone-bridge  Method  of  Measuring  a  Resistance  from  10  to 

1000  Megohms 153 

802.  Use  of  a  Capacity  in  Connection  with  a  Wheatstone  Bridge  for 

High-resistance  Measurements 155 

803.  Major  Cardew's  Electrometer  Method  of  Measuring  a  High  Re- 

sistance    156 

804.  The   Measurement  of  High  Resistances,   Unassociated  with  an 

Appreciable  Capacity;    Deflection  Methods.  . 157 

805.  The  Galvanometer  and  Accessory  Apparatus  for  High-resistance 

Measurement 157 

806.  Galvanometer  Shunts 157 

807.  The  Ayrton  or  Universal  Shunt '.  160 

808.  Galvanometer  Constant,  Obtained  by  Using  an  Ayrton  Shunt.  .  .  .  166 

809.  Insulation  Measurements  with  a  Galvanometer  and  an  Ayrton 

Shunt 167 

810.  Measurement  of  High  Resistances  by  Leakage  Methods 170 

811.  Theory  of  Leakage  of  Condensers 170 

812.  High  Resistance  Measured  by  Leakage;  Method  1 171 

813.  High  Resistance  Measured  by  Leakage;   Method  II 175 

814.  Insulation  Resistance  of  a  Celluloid  Condenser  Obtained  by  the 

Method  of  Leakage 176 

815.  High  Resistance  Measured  by  Leakage;  Method  III 180 


X  CONTENTS 

CHAPTER  IX.  —  INSULATION  RESISTANCE  OF  CABLES. 

ART.  PAGE 

900.  Introductory  Note 185 

901.  Formula  for  Calculating  the  Insulation  Resistance  of  a  Cable 185 

902.  Theorem    upon    the    General    Relation    Between    Capacity  and 

Resistance 186 

903.  Application  of  Theorem  to  the  Measurement  of  a  High  Resistance 

by  Leakage 189 

904.  Insulation  Resistance  of  a  Long  Cable  by  Deflection  Methods ....      191 

905.  Factory  Testing  Set  for  Insulation  Measurements 198 

CHAPTER  X.  —  RESISTANCE  AS  DETERMINED  WITH  ALTERNAT- 
ING CURRENT. 

1000.  Remarks  upon  Resistance  when  Determined  with  Alternating 

Current .,  .  .  .      199 

1001.  To    Measure    an    Alternating-current    Resistance;     Apparatus 

Required 200 

1002.  Description  of  Circuits  and  Theory  of  Method 201 

1003.  Directions  for  Using,  and  Test  of  Method 207 

CHAPTER  XI.  —  RESISTANCE  MEASUREMENTS  WHEN  THE 
RESISTANCE  INCLUDES  AN  ELECTROMOTIVE  FORCE. 

1100.  Material  Included  Under  this  Title 210 

1101.  Measurement  of  Insulation  of  an  Electric  Wiring  System  while  the 

Power  is  On .' .  .  .     210 

1102.  Voltmeter  Method  for  Insulation  Measurement  while  the  Power 

is  On.... 210 

1103.  Galvanometer  Method  for  Insulation  Measurement  while  Power 

is  On 212 

1104.  Determination  of  the  Internal  Resistance  of  Batteries 214 

1105.  Battery  Tests  by  Condenser  Method 215 

1106.  Mance's   Method  of  Measuring  the   Internal  Resistance  of  a 

Battery .  .  .  .' 218 

1107.  Voltmeter  and   Ammeter   Methods  of   Measuring   the   Internal 

Resistance  of  a  Battery 220 

1108.  A  Word  on  Polarities 221 

1109.  "Voltmeter  and  Ammeter  Method;    Principle  of  Polarities  Illus- 

trated      222 

1110.  Total  Resistance  of  a  Network  between  Two  Points  when  the 

Branches  of  the  Network  Contain  Unknown  E.M.F's 224 

1111.  Alternating-current  Methods  of  Measuring  the  Resistance  of  a 

Battery 226 

1112.  Bridge  Method;  Telephone  Detector 226 

1113.  Bridge  Method;  Electrodynamometer  Detector.  .  230 

1114.  Electrodynamometer  Substitution  Method  (Author's  Method).  .  231 

1115.  Galvanometer  Deflection  Methods  for  Obtaining  the  Resistance  of 

a  Battery. 233 


CONTENTS  XI 


AKT. 

1116.  Diminished  Deflection  Method  ..............................  233 

1117.  Kelvin's  Method  ..........................................  235 

1118.  Siemens'  Method  ..........................................  236 

1119.  Resistance  of  Electrolytes  ..................................  238 

1120.  The  Method  of  Kohlrausch  for  Measuring  the  Resistivity  of  an 

Electrolyte  ..........................  ....  240 

1121.  Determination  of  Relative  Resistivities  of  Electrolytes  .........  244 

1122.  Hering's  Liquid  Potentiometer  Method  for  Determining  Elec- 

trolytic Resistances  ....................................  247 

1123.  The  Substitution  Method  ...................................  248 

1124.  Resistance  of  "  Grounds"  (Bell  Telephone  Method)  ............  248 


CHAPTER  XII.  —  ELEMENTARY  PRINCIPLES  OF  FAULT  LOCATION. 

1200.  Fault  Location 251 

1201.  Faults  Occurring  on  Land  Lines 252 

1202.  Problems  in  Fault  Location 253 

1203.  Relation  of  Principles  to  Practice  in  Fault  Location 253 

1204.  Location  of  a  Ground  upon  a  Single  Line  with  Only  an  Earth 

Return 253 

1205.  Loop  Methods  for  Locating  Grounds  or  Crosses 258 

1206.  Notes  on  the  Varley  Test 265 

1207.  Modified  Loop  Methods  to  Meet  Special  Conditions 267 

1208.  Where  the  Faulty  Wire  is  of  Known  Length  and  there  is  Only 

One  Good  Wire  of  Unknown  Length  and  Resistance 271 

1209.  One  Good  Wire  of  Unknown  Length  and  Two  Faulty  Wires 

Equal  in  Length  and  Resistance 276 

1210.  Methods  of  Applying  Corrections  in  Loop  Tests 278 

1211.  Location  of  Grounds  on  High-tension  Cables 281 

1212.  Location  of  Faults  upon  Low-tension  Power  Cables 283 

1213.  Method  of  Locating  Grounds  upon  Heavy,  Short,  Underground 

Cables 284 

1214.  The  Location  of  Opens 284 

1215.  Location  of  Inductive  Crosses 288 

1216.  Comments  on  Practice  and  Accuracy  in  Fault  Location 290 

1217.  A  Word  on  Fault-locating  Apparatus 293 

CHAPTER  XIII.  —  MEASUREMENT  OF  TEMPERATURE  BY  THE 
MEASUREMENT  OF  RESISTANCE. 

1300.  Remarks  on  Temperature  and  Thermometry 296 

1301.  Electrical-resistance  Thermometry 297 

1302.  Construction  of  Resistance  Thermometers 302 

1303.  Methods  of  Reading  Resistance  Thermometers 308 

1304.  Slide-wire  Bridge  Method 308 

1305.  Differential-galvanometer  Method 312 

1306.  Resistance-thermometer  Bridge  with  Two  Traveling  Contacts.  . .  312 

1307.  Use  of  Dial  Bridges  for  Temperature  Measurements 314 


xii  CONTENTS 

ART.  PAGE 

1308.  Kelvin  Double-bridge  Method  of  Reading  Temperature 315 

1309.  Direct-deflection  Method  of  Reading  Temperatures 317 

1310.  Deflection  Methods;   Using  Constant  Currents 319 

1311.  The  Measurement  of  Extremely  High  Temperatures 322 


CHAPTER  XIV.  —  INSTRUMENTS  USED  FOR  MEASURING  RESISTANCE. 
SOME  GENERAL  PRINCIPLES  CONSIDERED. 

1400.  Proposed  Treatment  of  Subject  ..............................  324 

1401.  Conformity  in  the  Parts  of  an  Outfit  .........................  324 

1402.  Sensibility  and  Accuracy  ...................................  325 

1403.  Resistance  Standards  .......................................  326 

1404.  Resistance  Boxes  and  Wheatstone  Bridges;  General  Remarks.  .  .  .  331 

1405.  Watt  Capacity  of  Resistance  Units  .................  ..........  332 

1406.  Construction  of  Resistance  Spools  ...........................  333 

1407.  The  Precision  of  Coils  in  Resistance  Sets  .....................  334 

1408.  Some  Features  of  Outside  Construction  .......................  335 

CHAPTER  XV.  —  DEFLECTION  INSTRUMENTS  AND  GALVANOMETERS. 

1500.  Distinction  Between  Indicators  and  Deflection  Instruments  .....  338 

1501.  Pointer  Type,  Flat-coil  Galvanometers  .......................  340 

1502.  Sensitive  Galvanometers  for  Refined  Measurements  of  Resistance 

and  Insulation  Testing  .................................  346 

1503.  The  Equation  of  Motion  of  a  Galvanometer  System  ............  346 

1504.  Comparison  of  Galvanometers  ...............................  349 

1505.  Description  of  One  Type  of  High-sensibility  Galvanometer  .....  361 

APPENDIX. 
I.   TABLE. 

(1)  Values  of                   ....................................  363 


II.   MATHEMATICAL  QUANTITIES  AND  RELATIONS. 

(1)  Functions  of  IT  and  e  ....................  ...............  365 

(2)  English-Metric  and  Metric-English  Conversions  ..........  365 

(3)  Formulae  for  the  Conversion  of  Temperature  Scales  .......  366 

(4)  Formulae  for  Temperature  Coefficients  ...................  366 

(5)  Relations  Between  Resistance  and  Conductivity  ..........  367 

(6)  Conversions  from   Practical    to   Electrostatic,    to   Electro- 

magnetic Units.     (C.G.S.  System)  ....................  371 

(7)  Approximation  Formulae.     Certain  other  Expressions  ......  372 

III.   WIRE  DATA  AND  FORMULA. 

(1)  Wire  Table  ...........................................  374 

(2)  The  Ohm  ............................................  375 

(3)  Resistance  of  Wire  Wound  in  a  Channel  .................  375 

(4)  Certain  Formulae  for  Wire  ........  376 


CONTENTS  Xlll 

ABT.  PAQE 
IV.   PHYSICAL  DATA. 

(1)  Resistivity  of  Mercury .  376 

(2)  Resistivities  at  20°  C.     Densities  and  Melting  Points  of  the 

Solid  Elements 377 

(3)  Data  on  a  Few  Alloys 379 

(4)  Standard  Solutions  for  Calibrating  Purposes 379 

(5)  Resistivities  of  Insulators 38° 

INDEX..                                       381 


METHODS    OF    MEASURING 
ELECTRICAL  RESISTANCE 


CHAPTER  I. 

EXTENT,   CHARACTER,   AND  PRECISION  OF  ELEC- 
TRICAL MEASUREMENT.    THEORY  OF 
ERRORS.    OHMIC  RESISTANCE. 

100.  Electrical  Measurement.  —  The  quantities  to  measure 
and  the  methods  of  making  measurements  are  more  numerous  in 
electrical  science  than  in  any  other,  and  when  there  is  added  the 
special  tests  required  in  connection  with  industrial  practice,  the 
extent  of  the  subject  is  far  too  great  for  a  full  and  adequate  treat- 
ment in  a  single  volume.  The  author  has  chosen,  therefore,  to 
present  but  one  phase  of  the  general  subject  —  methods  of  meas- 
uring electrical  resistance.  We  shall  do  well,  however,  to  first 
consider  briefly  what  is  usually  comprehended  under  the  subject 
of  electrical  measurement,  the  nature  of  the  problems  involved 
and  some  of  the  fundamental  principles  which  pertain  to  all  kinds 
of  electrical  measurement.  This  will  give  a  clearer  understand- 
ing of  the  relation  which  methods  of  measuring  resistance  bear 
to  electrical  measurement  in  general. 

In  addition  to  several  more  or  less  fundamental  quantities, 
industrial  practice  requires  the  determination  or  measurement  of 
other  quantities,  such  as:  reactance;  impedance;  frequency  of  a 
periodically  varying  current  or  E.M.F. ;  phase  differences;  power 
factor;  location  of  short  circuits  in  coils;  ratios  of  transformers; 
location  of  faults  in  linear  conductors.  Very  many  special  deter- 
minations are  also  made,  such  as  the  calibration  of  instruments, 
the  testing  of  conductors,  conductor-insulation  and  all  kinds  of 
electric  power  machinery,  etc.  Special  methods  of  measurement 
have  been  devised  in  many  cases  to  meet  these  various  requirements. 

All  the  electrical  quantities  may  be  steady  or  constant  in  value, 
or  they  may  vary  in  a  determinate  manner.  In  the  latter  case 

1 


» 

2       «;' 


ELECTRICAL  RESISTANCE         [ART.  101 


the  methods  of  measurement  and  the  instruments  employed  are 
usually  quite  different  from  those  in  the  former  case.  Electrical 
measurements  are,  therefore,  ordinarily  considered  under  direct- 
current  measurements  and  alternating-current  measurements.  Cur- 
rents and  E.M.F.'s  which  vary  rapidly  in  an  entirely  indeter- 
minate manner  are  now  studied  and  measured  with  the  aid  of  the 
oscillograph. 

The    quantities    which    principally    require    consideration    in 
ordinary  industrial  electrical  measurements  are  the  following: 


Quantity 

Symbol 

Dimensions 
(El'  mag. 
System) 

Practical  units 

Quantity  of  electricity  
Potential  difference 

Q 
v 

L\U\ 

rfM'T-2 

Coulomb. 
Volt. 

Electromotive  force  
Electric  current  
Ohmic  resistance  
Resistivity  

E 
I 
R 

p 

L\M\T~* 
L>M*T~l 
LT-i 
L2T~l 

Volt. 
Ampere. 
Ohm. 
Ohm-centimeter. 

Conductance  

G 

L~1T 

Mho. 

Conductivity  .  . 

cr 

L~2T 

Mho  per  centimeter 

Capacity  .  . 

c 

L-1TZ 

Microfarad. 

Inductance  .  .  . 

L 

L 

Henry. 

Specific  inductive  capacity.  . 

Strength  of  magnetic  pole  ... 
Magnetic  induction  
Magnetizing  force  
Magnetic  permeability  

Electric  energy.  ... 

k 

m 
B 
H 

M 

w 

number 

L3  M1  T-1 
L-$M*T-i 
L~*M*T-i 
number 

L2MT~2 

No  name. 

No  name. 
Gauss. 
Gilbert  per  Cm. 
No  name. 

Joule. 

Electric  power  
Length  . 

p 

L 

L2MT~* 
L 

Watt. 
Centimeter 

Mass. 

M 

M 

Gram 

Time  

T 

T 

Second. 

101.  Some  General  Principles.  —  Measurement  always  in- 
volves the  process  of  finding,  by  direct  or  by  indirect  means,  how 
many  times  some  quantity,  which  we  choose  to  call  the  unit,  is 
contained  in  some  other  quantity  of  the  same  kind,  the  magni- 
tude of  which  we  wish  to  determine.  In  the  actual  process  of 
measurement,  the  quantity,  which  is  selected  as  the  unit,  must 
be  something  more  than  a  mere  abstraction  or  definition.  It 
must  be  represented  by  a  concrete  thing.  Thus  the  unit  of 
length,  in  the  metric  system,  is  defined  as  the  one  ten-millionth 
of  the  earth's  quadrant  and  is  called  the  meter,  but  the  real  unit 
of  length  is  an  actual  distance  between  two  marks  upon  a  par- 


ART.  101]      EXTENT  OF  ELECTRICAL   MEASUREMENT  3 

ticular  bar  of  metal.  The  entire  scientific  world  has  agreed  to 
call  the  distance  between  these  two  marks  the  real  unit  of  length, 
which  is  called  a  meter.  There  are  numberless,  more  or  less 
accurate  copies  of  this  standard  of  length,  and  whenever  an  actual 
length  measurement  is  made,  the  procedure  consists  essentially 
in  finding  how  many  times  the  length  of  one  of  these  copies  of 
the  standard  meter  is  contained  in  the  length  being  measured. 
The  concrete  thing  which  embodies  and  represents  the  unit  of 
definition  is  generally  and  properly  called  a  standard.  The  stand- 
ard may  represent  the  unit  exactly,  as  in  the  case  of  the  meter, 
or  it  may  be  a  known  multiple  or  fraction  of  the  unit  repre- 
sented. Thus  the  column  of  pure  mercury  of  uniform  cross-sec- 
tion which  is  106.3  centimeters  long  and  has  a  mass  of  14.4521 
grams,  or  a  cross-section  of  1  mm2  is,  when  at  a  temperature  of 
0°  C.,  an  exact  concrete  realization  of  the  unit  of  resistance, 
called  the  ohm.  A  standard  cadmium  cell,  on  the  other  hand, 
which  bears  a  certified  value  of  1.0183  volts  is  a  concrete  reali- 
zation which  represents  a  known  multiple  of  the  unit  of  electro- 
motive force. 

It  is  desirable  and  convenient  that  a  standard  should  represent 
the  unit  exactly  or  some  simple  even  multiple  of  it,  but  the  diffi- 
culties in  the  way  of  accomplishing  this  are  often  great,  as  in  the 
case  of  the  standard  of  E.M.F.  just  cited.  For  exact  work  it  is 
customary  to  construct  standards  to  equal  the  unit  or  its  even 
multiple  as  nearly  as  possible,  and  furnish  certificates  giving  the 
exact  value  in  terms  of  the  unit  represented.  Thus  a  standard 
one-microfarad  condenser  is  seldom  constructed  to  equal  one 
microfarad  closer  than  one-tenth  or  one-quarter  of  one  per  cent, 
tho  condensers  can  be  compared  with  one  another  much  closer 
than  this.  The  value  stamped  upon  the  standard  is  called  the 
nominal  value,  and  the  certificate  which  goes  with  it  states  the 
amount,  generally,  or  at  least  preferably,  in  percentage  by  which 
it  is  greater  or  less  than  its  nominal  value. 

When  a  measurement  is  made,  it  is  determined  by  some 
selected  method  of  procedure  how  many  times  the  magnitude  of 
the  standard  is  contained  in  the  quantity  measured.  If  this  quan- 
tity is  smaller  than  the  standard,  then,  of  course,  it  will  be  found 
to  contain  the  standard  a  fractional  number  of  times. 

The  measurement  may  be  in  error  from  two  causes ;  either  the 
standard  may  be  larger  or  smaller  than  it  is  certified,  in  which 


4  MEASURING  ELECTRICAL   RESISTANCE         [ART.  102 

case  the  quantity  being  measured  will  be  called  smaller  or  largei 
than  it  really  is,  or  the  process  of  finding  how  many  times  the 
quantity  measured  contains  the  magnitude  of  the  standard  may 
be  incorrectly  applied.  The  responsibility  of  the  first  error  rests 
with  the  standard  and  of  the  second  error  with  the  user  of  the 
standard. 

In  industrial  measurements,  the  one  engaged  in  making  the 
measurements,  that  is,  in  finding  how  many  times  the  magnitude 
of  the  standard  he  employs  is  contained  in  the  quantity  he  is 
measuring,  seldom  undertakes  a  test  of  the  precision  of  his  stand- 
ards. He  relies  upon  others  for  this,  and  herein  rests  the  vast 
importance  and  responsibility  of  a  Standards'  Bureau  like  the  one 
we  have  in  Washington.  It  can  be  shown  that,  in  a  final  analysis, 
the  accuracy  of  most  of  the  measurements  made  in  America  rests 
upon  the  facilities,  the  intelligence,  and  conscientious  care  of  this 
Bureau. 

102.  Comments  on  Accuracy  and  Method.  —  The  precision 
with  which  any  measurement  is  made  may  vary  all  the  way  from 
a  rough  estimation  to  the  high  refinement  attained  by  a  prolonged 
investigation  where  the  total  error  may  be  reduced  to  a  few  parts 
in  a  million.  This  being  so,  the  prominent  question  to  beheld 
before  the  mind,  when  starting  a  series  of  measurements,  should 
be:  Under  all  the  circumstances  of  the  case,  what  degree  of  pre- 
cision is  one  justified  in  seeking?  Assuming  the  accuracy  of  the 
standards  the  possible  precision  attainable  is  generally  related  very 
closely  to  the  time  spent  upon  the  measurements,  and  the  time 
which  one  is  justified  in  using  is  governed  by  circumstances  which 
should  not  be  ignored.  Suppose,  for  example,  the  object  of  a 
measurement  is  to  determine  the  specific  resistance  of  a  sample 
of  commercial  graphite,  With  accurate  standards  and  the  expen- 
diture of  much  time  this  might  be  determined  to  an  accuracy, 
perhaps,  of  a  twenty-fifth  of  one  per  cent.  But  there  would  be 
no  justification  in  seeking  such  a  high  precision  in  this  case 
because  it  would  be  without  value  in  view  of  the  variability  of 
graphite.  It  is  probable  that  different  samples  of  graphite  from 
the  same  supply  would  differ  in  resistivity  by  one  per  cent  or 
more,  and  the  same  sample  would  certainly  vary  in  resistivity  from 
day  to  day  by  much  more  than  a  twenty-fifth  of  one  per  cent.  On 
the  other  hand,  if  the  object  of  a  series  of  measurements  is  to 
determine  the  minute  variations  over  a  period  of  time  which  take 


ART.  102]      EXTENT  OF  ELECTRICAL   MEASUREMENT  5 

place  in  a  manganin-resistance  standard,  the  most  painstaking 
care  should  be  exercised  to  secure  the  necessary  precision,  and  this 
care  and  the  time  spent  would  be  entirely  justified,  if  circumstances 
justified  the  research  itself. 

Again,  great  accuracy  and  painstaking  care  in  finding  how  many 
times  the  magnitude  of  the  standard  available  is  contained  in  the 
quantity  under  measurement  is  not  justified,  if  the  uncertainty 
must  remain  great  regarding  the  true  value  of  the  standard  itself. 
Few  lines  of  work  require,  as  does  electrical  measurement,  such 
discriminating  judgment  as  to  the  relative  importance  of  things. 
But  no  admonition  or  instruction  can  give  this  balanced  judgment, 
so  necessary  to  successful  performance,  as  does  practice.  In  this, 
as  in  most  matters  requiring  skill,  to  measure  well,  one  must 
practice  measurement.  When,  by  experience,  a  discriminating 
judgment  has  been  acquired,  it  will  be  applied,  not  consciously,  but 
quite  instinctively. 

In  making  electrical  measurements  much  more  mischief  is  likely 
to  result  from  a  careless  commitment  of  gross  errors  than  from 
failure  to  give  attention  to  details  and  to  deduce  the  most  prob- 
able value  from  a  set  of  observations.  The  gross  errors  may 
result  from  a  misconception  of  the  nature  of  the  problem,  from  an 
entirely  incorrect  reading  of  the  larger  indications  of  an  instrument, 
from  mistakes  made  in  ordinary  arithmetic,  or  from  an  improper 
interpretation  of  units  in  calculating  the  results.  The  strained 
attention  often  required  to  read  small  decimals,  frequently  causes 
an  entire  loss  of  mental  perspective  as  to  the  main  features  of  the 
problem.  Important  points  are  overlooked  while  minutiae  are 
carefully  observed.  In  no  line  of  effort  is  a  novice  more  apt  "to 
strain  at  a  gnat  and  swallow  a  camel,"  than  when  trying  to  make 
a  refined  electrical  measurement. 

It  is  generally  wise,  as  a  precautionary  measure,  after  the 
apparatus  is  in  adjustment  and  all  the  connections  are  completed, 
to  make  what  might  be  termed  a  survey  measurement,  and  then 
to  deduce  the  results  with  a  rough  calculation,  and  consider  well 
if  these  results  look  reasonable.  If  they  do,  and  the  outfit  is 
seen  to  be  in  a  proper,  balanced,  working  condition,  painstaking 
observations  may  then  be  undertaken  and  the  data  worked  up. 
Nothing  so  assists  in  proving  or  disproving  the  reasonableness  of 
the  observations  as  plotting  the  data  in  a  curve,  and  this,  in 
almost  every  case,  is  to  be  recommended. 


6  MEASURING  ELECTRICAL  RESISTANCE        [ART.  102 

In  nearly  every  extended  series  of  observations  there  will  be, 
in  addition  to  accidental  errors  which  are  as  likely  to  be  plus 
as  minus,  certain  systematic  errors  which  escape  observation  and 
are  not  eliminated  by  taking  the  mean  value  of  a  number  of  read- 
ings. The  novice,  noting  a  fine  agreement  among  his  observa- 
tions, is  often  deceived  into  supposing  he  has  attained  an  accuracy 
far  greater  than  the  measurements  really  justify.  The  surest 
way  to  obtain  enlightenment  is  to  remeasure  the  same  quantity 
by  an  entirely  different  method.  The  lack  of  agreement  that  is 
apt  to  result  is  often  a  disagreeable  surprise  and  gives  one,  as 
nothing  else  can,  a  just  estimate  of  how  grudgingly  Nature  per- 
mits the  real  truth  to  be  extracted.  A  close  agreement -in  the 
results  of  measurements  made  by  two  or  three  entirely  different 
methods  gives,  on  the  other  hand,  the  highest  assurance  that  a 
real  precision  has  been  attained. 

When  a  quantity  is  to  be  measured  with  precision,  it  is  generally 
wise  to  seek  a  method  of  measurement  which  will  give  the  result 
as  directly  as  possible  and  without  the  necessity  of  making  a 
number  of  corrections.  So-called  "null"  methods,  in  which  the 
quantity  being  measured  is  balanced  against  some  other  quantity, 
are  less  rapid,  as  a  rule,  than  deflection  methods,  in  which  the 
quantity  is  measured  in  terms  of  the  deflection  of  some  instru- 
ment, but  ordinarily  the  former  are  much  more  accurate  and  there 
is  less  necessity  of  applying  corrections.  Null  methods  are  gener- 
ally to  be  preferred  for  all  precision  work.  In  the  descriptions 
which  follow,  of  the  various  methods  of  measuring  electrical  re- 
sistance, the  relative  advantages  of  the  two  methods  will  become 
obvious. 

The  question  as  to  what  sensibility  the  indicating  or  measuring 
instrument  should  have  is  an  important  one  that  must  receive 
careful  consideration.  It  may  be  said,  however,  that,  in  general, 
increased  sensibility  in  the  indicating  instrument  leads  to  a  re- 
duction in  size  and  cost  of  all  the  rest  of  the  equipment  required. 
On  the  other  hand,  more  care,  time,  and  skill  are  required  to 
work  with  sensitive  instruments,  and  judgment  must  be  con- 
stantly exercised  to  choose  a  sensibility  best  adapted  to  the  particu- 
lar problem  in  hand. 

It  should  be  borne  in  mind  that  the  only  electrical  quantity 
which  can  be  sold  or  exchanged  for  money  is  electrical  energy, 
or  electrical  power  expended  for  a  given  time.  Electrical  power 


ART.  103]      EXTENT   OF  ELECTRICAL  MEASUREMENT  7 

is  composite,  being  the  product  of  E.M.F.  and  the  current  which 
is  in  phase  with  the  E.M.F.  When  the  average  power  developed 
in  a  certain  time  is  multiplied  by  the  time,  the  total  amount  of 
electrical  energy  developed  is  obtained.  Now  energy  has  a  real 
existence  and  a  market  value.  Consequently,  from  the  industrial 
standpoint,  most  electrical  measurements  have  as  their  final 
object  the  precise  determination  of  that  which  has  money  value, 
namely,  electrical  energy  or  watt-hours.  While  there  are  both 
scientific  and  practical  considerations  which  make  it  necessary 
to  measure  separately  such  quantities  as  E.M.F.,  current,  phase 
angles,  resistances,  etc.,  one  should  not  lose  sight  of  the  industrial 
object  to  be  attained,  which  is  the  proper  and  just  charge  for  a 
quantity  of  electrical  energy  sold.  Other  considerations,  of  course, 
apply  with  those  classes  of  measurement  required  in  telephony 
and  telegraphy,  or  those  made  solely  for  scientific  investigation. 

Finally,  it  is  strongly  recommended  to  all  those  engaged  in 
electrical  measurements  that  they  give  a  careful  study  to  the 
meanings  and  physical  interpretations  of  the  electrical  and  mag- 
netic units  in  use.  In  this  connection  the  writer  would  recom- 
mend the  use  and  study  of  a  little  book  called  "  Conversion 
Tables,"  by  Dr.  Carl  Hering. 

103.  Elements  of  the  Theory  of  Errors.*  —  There  are,  in  gen- 
eral, two  classes  of  errors;  systematic  errors  and  accidental  errors. 
The  former  class  often  result  from  an  incorrect  value  being  as- 
signed to  the  standards  employed,  from  a  faulty  calibration  of 
the  scale  of  an  instrument,  or  from  the  constant  presence  and 
influence  of  an  unrecognized  force,  as,  for  example,  the  unrecog- 
nized presence  of  a  thermoelectric  force  in  the  circuit  of  the  indi- 
cating galvanometer.  Systematic  errors  do  not  eliminate  when 
the  mean  value  of  a  series  of  observations  is  taken  as  the  result. 

The  latter  class  generally  result  from  inaccuracies  in  reading 
the  instruments  and  from  fluctuations  above  and  below  a  mean 
value  of  some  quantity  which  determines  the  readings  of  the 
instruments  while  the  observations  are  being  taken;  for  example, 
fluctuations  in  the  E.M.F.  employed  when  measuring  resistances 
with  deflection  instruments.  As  accidental  errors  are  as  likely 
to  be  plus  as  minus,  they  tend  to  eliminate  from  the  mean  value 
as  the  number  of  readings  is  increased.  But  it  may  be  added, 

*  Some  of  what  follows  under  this  heading  is  taken  from  "Instruments  et 
Methods  de  Mesures  electriques  Industrielles,"  par  H.  Armagnat. 


8 


MEASURING  ELECTRICAL  RESISTANCE       [ART.    103 


that  the  theory  of  probability  shows  that  the  precision  increases 
not  directly  with  the  number,  but  proportionally  with  the  square  root 
of  the  number  of  measurements. 

The  difference  between  the  value  found  in  a  single  measure- 
ment and  the  mean  of  the  entire  series  of  measurements  is  called 
the  apparent  error.  If  the  sum  of  all  the  apparent  errors,  without 
regard  to  sign,  be  taken,  and  this  sum  be  divided  by  the  number 
of  measurements,  we  obtain  the  mean  error.  The  theory  of 
probability  shows  that  the  mean  error,  obtained  in  this  way, 
provided  the  errors  are  accidental  and  not  systematic,  is  very 
closely  the  same  as  would  be  obtained  if  the  mean  error  were 
found  by  taking  the  true  value  of  the  quantity  instead  of  the 
mean  value.  Hence,  in  estimating  the  true  value  from  a  set  of 
measurements,  one  can  say  that  this  true  value  is  equal  to  the 
mean  value  found  by  the  measurements  plus  or  minus  the  mean 
error.  For  example,  suppose  one  has  made  five  measurements  of 
a  resistance,  with  the  same  apparatus  and  method,  and  all  pre- 
cautions have  been  taken  to  avoid  systematic  errors;  the  results 
would  be  expressed  as  follows: 


No  of  Meas. 

Ohms  found 

Apparent  error 

1 
2 
3 
4 
5 

25.6 
25.3 
25  7 
25.5 
25  7 

+0.04 
-0.26 
+0.14 
-0.06 
+0.14 

127.8  =  sum 

0.640  =  sum 

25  .  56  =  mean 

0.  128  =  mean  error 

True  value  =  25.56  ±  0.128,  which  one  would  call 
25.56  =t  0.13. 

It  is  rare  in  electrical  measurements  that  there  is  any  need  to 
apply  the  more  exact  methods  for  arriving  at  the  most  probably 
true  result,  which  the  theory  of  probabilities  teaches  us  to  apply, 
and  the  matter  will  not  be  further  considered  here. 

The  numerical  difference  between  the  result  of  a  measurement 
and  the  true  value  of  a  quantity  measured  is  called  the  absolute 
error.  The  ratio  of  the  absolute  error  to  the  magnitude  measured 
is  called  the  relative  error.  One  must  take,  however,  in  place  of 
the  true  value,  which  is  unknown,  the  mean  value,  for  expressing 


ART.  103]      EXTENT  OF  ELECTRICAL   MEASUREMENT  9 

the  relative  error.     Thus,  in  the  example  above,  ±  0.128  is  the 

0  1 28 

mean  absolute  error,  and  ±  ^~r^  =  ±  0.00500  is  the  mean  relative 

^o.ou 

error. 

If  the  relative  error  is  multiplied  by  100,  it  is  then  called  the 
per  cent  error.  Thus,  in  the  above  example,  ±  0.00500  X  100 
=  =b  0.500  of  1  per  cent.  Namely,  the  value  obtained  is  equal  to 
25.56  ohms  with  a  probable  error  of  plus  or  minus  one-half  of 
one  per  cent. 

The  relative  error  or  the  per  cent  error  is  the  error  which  is  of 
interest,  because  the  absolute  error  must  always  be  considered 
in  relation  to  the  absolute  value,  if  it  is  to  have  any  physical 
meaning.  Thus  to  measure  1000  ohms  with  a  plus  or  minus  error 
of  1  ohm  is  quite  permissible  but  to  measure  10  ohms  with  a  plus 
or  minus  error  of  1  ohm  would  be  but  rough  work.  In  the  former 
case  the  precision  would  be  0.1  of  1  per  cent  and  in  the  latter  case 
but  10  per  cent. 

In  all  that  follows,  unless  specifically  stated  to  the  contrary,  we 
shall,  in  speaking  of  errors,  always  refer  to  the  relative  error,  or  to 
the  per  cent  error. 

The  result  of  a  measurement  is  often  given  so  as  to  be  dependent 
upon  several  partial  results,  or  separate  measurements. 

Let  x  represent  the  value  sought,  and  y  the  phenomenon,  as  for 
example  the  deflection  of  a  voltmeter,  upon  which  the  value 
depends,  then  x  will  be  some  function  of  y,  or 

x  =  F  (y).  (1) 

If  an  absolute  error  A?/  is  made  in  observing  the  phenomenon,  the 
result  will  be  in  error  some  amount  A#,  such  that 

x  +  Az  =  F  (y  +  Ay),  (2) 

or 

A*  =  F  (y  +  Ay)  -  F  (y).  (3) 

If  the  error  made  in  y  is  small,  we  can  treat  the  increment  in  y  as  a 
differential,  in  which  case  we  can  write 

dx  =  F'  (y)  dy,  (4) 

and  the  relative  error  will  then  be 

dx  _  F'  (y)  dy  ( 

x~       F(y) 

To  illustrate  Eq.  (5);  suppose  it  is  required  to  determine,  by  meas- 
uring a  current,  the  rise  in  the  temperature  T  of  a  conductor,  caused 
by  the  current  7  which  it  is  made  to  carry.  If  we  assume  the 


10  MEASURING  ELECTRICAL  RESISTANCE         [ART.  103 

rise  of  temperature  of  the  conductor  to  vary  as  the  square  of  the 
current  carried,  we  can  write 

T  =  KI2,  where  K  is  a  constant. 
We  then  have 

x  is  equivalent  to  T,  y  is  equivalent  to  7, 

F  (y)  is  equivalent  to  KI2,  dx  is  equivalent  to  dT, 

and 

Ff  (y)  dy  is  equivalent  to  2  KI  dl. 

Hence,  by  Eq.  (5), 

dT  =  2KIdI  ^2dl 
T  =       KI2  I 

Eq.  (6)  shows  that  the  relative  error  in  the  determination  of  the 
rise  in  temperature  will  be  twice  as  great  as  the  relative  error 
made  in  measuring  the  current. 

Eq.  (5)  may  be  extended  to  the  case  of  several  variables,  and 
so  permit  us  to  estimate  in  advance  the  relative  precision  of  the 
final  result,  when  we  know  the  magnitude  of  the  errors  that  may 
be  made  in  each  of  the  separate  elements  measured  and  upon 
which  the  final  result  depends. 

Let 

x  =  F  (u,  v,  Wj  etc.).  (7) 

Differentiate  this  function  in  respect  to  each  of  the  variables  u, 

v,  Wj  etc.     Then  we  derive 

dx_F'u(u,  v,w,...)  du+F'v  (u,  v,  w,  .  .  .  )  dv+F'w(u,  v,  w)  dw-\-  •  •  • 


x  F  (u,v,w,  .  .  .  ) 

(8) 

Apply  Eq.  (8)  to  the  case  of  a  circuit  in  which  we  wish  to  measure 
the  power  consumed.  If  P  is  this  power,  R  the  resistance  of  the 
circuit,  and  7  the  current  flowing,  we  have 

P  =  RI2.  (9) 

If  to  determine  P  we  have  to  measure  R  and  7,  we  shall  have 

P  =  F(R)I)') 
hence,  by  Eq.  (8), 

dP  _  R2IdI  +  I2dR  _  2dl      dR 
=          ~~  7      !"'•«' 


Now,  -y-  is  the  relative  error  in  measuring  the  current,  which  call 

EJ,  and  -pr  is  the  relative  error  in  measuring  the  resistance,  which 
JK 


ART.  103]      EXTENT   OF  ELECTRICAL   MEASUREMENT  11 

call  ER.     Then 

j?-  =  2EI  +  ER.  (11) 

Eq.  (11)  shows  that,  if  +  0.33 J  is  the  per  cent  error  made  in 
measuring  the  current  and  +  0.33J  is  the  per  cent  error  in  measur- 
ing the  resistance,  the  per  cent  error  in  determining  the  power 

will  be 

dP 
W0~  =  2  X  0.33  J  +  0.33 }  =  1  per  cent. 

If  the  per  cent  error  in  measuring  the  resistance  happened  to  be 
-  0.33J  then  the  per  cent  error  in  determining  the  power  would  be 
only  +  0.33 J.  But,  as  it  is  unknown  whether  the  accidental  errors 
are  positive  or  negative,  no  greater  precision  can  be  assigned  to 
the  result  than  would  be  deduced  upon  the  assumption  that  all 
the  partial  errors  have  a  like  sign. 

Certain  further  conclusions  can  be  drawn  from  Eqs.  (5)  and 
(8): 

1st.  The  quantity  Qa  being  measured  is  the  sum  of  two  factors 
x,  y,  or 

Q8  =  x  +  y. 

By  Eq.  (8)  the  relative  error  would  be 

dQs  _  dx  +  dy 

QT  =    ^TF 

It  is  to  be  noted  here  that  the  relative  error  in  the  result  cannot 
exceed  the  relative  error  committed  in  either  factor;  for  assume 

dQs      dx 
y  =  0,     then   -77-  =  — 

Vs  X 

2d.     The  quantity  Qd  is  the  difference  of  two  factors  x,  y,  or 

Qd  =  x  -  y. 
In  this  case  the  relative  error  becomes 

dQd  _dx-  dy  (     . 

Qd  =     x-y    ' 

Here  it  is  seen  that  the  relative  error  is  as  much  greater  as  the 
difference  (x  —  y)  in  the  two  quantities  is  smaller.  This  is  why, 
in  measuring  a  quantity  which  involves  the  difference  of  two 
factors,  one  will  always  obtain  a  result  which  is  much  less  exact 
than  is  obtained  in  measuring  each  of  the  elements. 


12  MEASURING  ELECTRICAL  RESISTANCE         [ART.  104 

3d.     The  quantity  Qp  is  the  product  of  two  factors,  or 

QP  =  xy. 

In  this  case  the  relative  error  will  be 
dQr      dx      dy 
QP        x        y 

Thus  the  relative  error  in  the  result  is  equal  to  the  sum  of  the 
relative  errors  committed  in  measuring  each  of  the  factors. 
4th.     The  quantity  Qq  is  the  quotient  of  two  factors,  or 

*-!• 

In  this  case 

ydx  _  xdy 
dQQ  _  j£ f_  _  dx      dy 

07s     ~T~      7  "7." 

y 

Hence,  in  this  case,  the  final  relative  error  is  equal  to  the  sum  of 
the  relative  errors  made  in  each  factor  when  these  have  opposite 
signs.  But  as  it  cannot  be  told  what  the  signs  will  be,  one  must 
assume  that  the  case  may  occur  in  which  the  signs  are  unlike. 

5th.     When  Q'p  is  the  power  m  of  a  factor  x,  as  Q'p  =  xm,  the 
relative  error  will  be 

dW»  =  mx^dx  =  mdx, 
QP  xm  x 

Hence  the  relative  error  in  the  result  will  be  m  times  as  great  as 
the  relative  error  made  in  the  factor. 

6th.     When  Qr  is  the  mth  root  of  the  factor  x,  as 

i 

Qr   =   Xm, 

the  relative  error  will  be 


Qr  ~  1         ~  m  x 

xm 

Hence,  the  relative  error  in  the  result  will  be  —  of  that  made  in 

m 

the  factor. 

104.   Application  of  the  Theory  of  Errors.  —  Having  in  mind 
the  above  principles,  we  can  reach  a  just  estimate  of  the  precision 


ART.  1041      EXTENT  OF  ELECTRICAL   MEASUREMENT  13 

which  may  be  expected  in  the  measurement  of  resistance  by  de- 
flection methods. 

In  practice,  the  instruments  most  used  are  such  as  have  a  scale, 
like  that  of  a  Weston  voltmeter  —  that  is,  a  scale  with  a  total  of 
150  divisions.  While  an  attempt  is  often  made  to  estimate  the 
readings  to  one-tenth  of  a  division,  it  is  thought  that  one-fifth 
of  a  division  is  as  close  an  estimation  as  can  be  relied  upon. 
Assuming  then  that  the  quantity  being  measured  is  given  directly 
by  a  full  scale  deflection,  'the  per  cent  error  can  scarcely  be  less 

than  ^-^ =  100  =  0.13  +  per   cent,   and  this   per  cent  error 

150  X  5 

steadily  increases  as  the  deflection  read  becomes  smaller.  With  a 
reading  of  ten  divisions  the  probable  error  would  be  2  per  cent. 
An  additional  source  of  error  will  be  introduced  if  the  scale  is  not 
laid  off  so  that  the  deflections  indicated  are  proportional  to  the 
current  passing  thru  the  instrument.  But  in  measuring  resistances 
with  Weston  voltmeters  and  millivoltmeters  this  source  of  error 
can  usually  be  disregarded.  Not  so,  however,  when  the  deflection 
instrument  is  a  galvanometer  with  telescope  and  scale  or  lamp 
and  scale.  In  such  case  the  scale  will  probably  have  250  divisions, 
each  side  of  a  central  zero,  and  one  might  be  led  to  expect  higher 
precision  than  may  be  obtained  with  a  pointer  instrument  of  only 
150  divisions.  The  scales  of  galvanometers,  however,  have  divi- 
sions of  uniform  length,  and  as  few  galvanometers  deflect  exactly 
proportional  to  the  current  thru  them,  the  deflections  indicated 
are  not  apt  to  be  accurately  proportional.  The  advantage 
therefore  of  a  longer  scale  may  be  offset  by  lack  of  proportionality 
in  the  deflections. 

Further,  in  resistance  measurements  by  deflection  methods,  the 
results  in  most  cases  are  not  given  directly  in  terms  of  a  single 
deflection  but  involve  the  difference  of  two  deflections,  and,  as 
appears  under  case  2  (§  103),  the  precision  of  the  final  result  will 
be  much  less  than  the  precision  with  which  the  individual  deflec- 
tions are  read. 

Again,  the  two  or  three  readings  which  must  be  taken  are  not 
made  simultaneously  but  in  succession,  and  this  procedure  always 
involves  the  assumption  that  all  conditions  influencing  the  pre- 
cision remain  constant  while  the  various  readings  are  being  taken. 
With  a  generator,  subject  to  variations  in  speed,  as  the  source 
of  current,  this  assumption  is  hardly  tenable. 


14  MEASURING  ELECTRICAL  RESISTANCE         [ART.  104 

The  theoretical  per  cent  error  in  any  of  the  cases  given  in  the 
following  chapter  for  measuring  resistances  is  easily  deduced  by 
making  use  of  the  principle  expressed  by  Eq.  (8)  (§  103).  We 
proceed  to  apply  the  principle  to  the  method  given  in  par.  207, 
where  two  unknown  resistances  x\  and  x2  in  series  are  to  be  deter- 
mined by  three  readings  of  a  deflection  instrument.  Let  one  of 
the  resistances  be  given  by  the  expression 


If  the  resistance  R  is  given  and  is  not  subject  to  variation,  the 
three  quantities  which  may  vary  and  which  are  not  independent 
are  D,  d\  and  d2.  If  we  apply  to  the  above  equation  the  operation 
expressed  by  Eq.  (8)  (§  103),  we  obtain 

dx±=     dD  -  ddi  dd2  (D  -  di) 

X!       D-di-dz      d2  (D-di-  d2)  ' 
Here  dxi,  dD,  ddi,  dd2  are  absolute  errors. 

These  errors  may  assume  either  a  positive  or.  a  negative  sign, 

hence  the  relative  error  -  -  will  depend  not  only  upon  the  magni- 

Xi 

tude  of  the  errors  dD,  dd\,  dd2  but  also  upon  their  sign.  If  the 
error  ddi  'is  equal  to,  and  of  the  same  sign  as  dD,  the  first  term  of 
Eq.  (2)  disappears. 

It  is  essential,  however,  in  computing  the  value  of  the  relative 
error  in  the  result,  to  assume  that  the  errors  made  in  the  elements 
(namely,  in  this  case,  in  D,  d\,  d2)  have  signs  which  are  the  least 
favorable  to  precision.  Assume,  then,  that  the  error  in  di  is 
negative. 

Now,  in  reading  a  voltmeter,  it  may  be  assumed  that  the 
conditions  would  be  so  arranged  that  the  deflection  D  comes  near 
the  top  of  the  scale,  and  it  may  be  assumed  that  the  readings  can 

be  made  to  -  of  the  largest  deflection,  or  to  —  .     Then  the  errors 

dD,  ddi,  dd2  may  all  be  as  large  as  —  and  most  unfavorable  to 

precision  as  far  as  sign  is  concerned.  With  these  assumptions  we 
shall  have 

D  +  D  D 

&,          p  +  p  p  (U      a>>  D   2d,  +  D-d, 

--  -- 


ABT.  104]      EXTENT  OF  ELECTRICAL   MEASUREMENT  15 

To  illustrate  Eq.  (3)  let  us  make  use  of  the  readings  obtained  in 
the  measurement  given  in  par.  207. 

r\          -I  rjrv  o 

Assume  that  —  =  -^~  ,  namely,  that  the  readings  can  be  made 
p 


to  -5^  of  the  greatest  reading.  Take  the  values  D  =  100.8, 
di  =  72.2,  dz  =  14.4.  Then  we  have,  by  Eq.  (3),  as  the  maximum 
per  cent  error, 

<fa,  100.8          2  X  14.4  +  100.8  -  72.2 

°^~          500  X  14.4  >       100.8  -  72.2  -  14.4 
=  ±  5.6  per  cent. 

The  per  cent  error,  actually  made,  was  only  +0.88  per  cent. 
This  may  be  due  in  part  to  a  more  accurate  reading  of  the  instru- 
ment than  was  assumed,  and  also  to  the  fact  that  the  errors  can- 
celled each  other;  if  the  latter  only  were  the  case  we  should  have 

100  ^  =  ±  -§-  nD~dl   ,  100  =  ±  2.8  per  cent. 
Xi  pdz  D  —  di  —  d2 

This  last  result  shows  that  in  the  particular  case  of  this  measure- 
ment the  readings  must  have  been  made  within  about  0.05  of  a 
scale-division. 

Returning  to  Eq.  (1)  (§  104)  we  see  from  par.  207  that,  if  one  of 
the  resistances  x2  is  infinity,  the  reading  di  would  be  zero  and  we 
have 


This  is  similar  to  the  case  given  in  Eq.  (8),  par.  201. 
If  now  in  Eq.   (2)  we  put  di  =  0,  we  shall  also  have  dd\  —  0, 
whence, 

dxi  _      dD  ddzD        _  dDdz  -  dd2D 

~zT  ~~  D  -  d2  ~  dz  (D  -  dz)  ~     dz  (D  -  d2)   ' 

=  ddz  =  ±  —  ,  we 
minus  sign  and  dD  with  a  plus  sign, 


If,  as  before,  dD  =  ddz  =  ±  —  ,  we  have,  by  taking  ddz  with  a 


m   *  =  100.  (4) 

Xi        pdz  D  —  dz 

Eq.  (4)  expresses  the  theoretical  maximum  per  cent  error  which 
would  be  obtained  in  the  method  described  in  par.  201  if  the 

readings  can  be  made  to  -  of  the  larger  deflection.    If  Eq.  (4)  be 


16  MEASURING  ELECTRICAL  RESISTANCE         [ART.  105 

applied  to  the  measurement  of  100  ohms,  as  given  in  the  table  in 

par.  205,  and  it  be  assumed  that  —  =  >  we  obtain  as  the  maxi- 

'p        J.UUU 

mum  per  cent  error  1.4  per  cent,  while  the  actual  error  was  0.76 
per  cent. 

Enough  has  now  been  given  to  show  the  low  accuracy  that  may 
be  expected  from  deflection  methods  of  measuring  resistances. 
These  methods,  nevertheless,  are  of  value  in  many  situations. 

If  it  be  remembered  that  all  copper  and  aluminum  conductors, 
such  as  magnet  coils  in  arc  lamps,  etc.,  change  in  resistance  about 
4  per  cent  for  every  change  of  10°  C.  in  temperature,  it  will  be 
recognized  that  the  rough  methods  of  measuring  resistances  with 
deflection  instruments  are  often  quite  as  precise  as  the  conditions 
demand,  and  because  of  their  simplicity  often  more  satisfactory 
than  methods  which  are  more  refined. 

The  apparatus  required  is,  furthermore,  very  generally  avail- 
able; the  measurements  may  be  made  quickly  and  in  places  where 
more  delicate  apparatus  could  not  well  be  used,  and  altho  the 
precision  attainable  is  low,  it  is  sufficient  to  meet  such  require- 
ments as  arise  in  connection  with  the  measurement  of  insulation 
resistances  and  the  resistances  of  copper  conductors,  such  as 
dynamo-field  coils. 

105.  Comments  on  Ohmic  Resistance.  —  An  ohmic  resist- 
ance, considered  as  a  quantity  to  be  measured,  may  be  viewed  in 
two  ways.  If  we  call  E  the  drop  of  potential  between  two  points 
in  an  electric  circuit,  and  /  the  current  flowing,  we  may  write 
either 

E  =  RI  (1) 

or 

E 
Ri  =  j-  (2) 

In  the  first  relation  the  drop  of  potential  is  expressed  as  propor- 
tional to  the  current  where  R  is  the  constant  of  proportionality. 
From  this  viewpoint,  R  is  considered  a  constant  coefficient  which 
pertains  to  a  particular  circuit  and  which,  when  multiplied  into 
the  current,  will  give  the  potential  drop.  The  relation  does  not 
assume  that  R  is  in  any  sense  a  function  of  7.  If  a  resistance,  so 
viewed,  is  to  have  an  exact  meaning,  it  must  possess,  under  con- 
ditions which  are  simple  to  state,  a  definite  value.  In  the  second 
relation,  R  i  is  viewed  as  a  numerical  quantity  which  merely 


ART.  1051      EXTENT   OF  ELECTRICAL   MEASUREMENT  17 

expresses  the  ratio  of  the  actual  potential-drop  between  any  two 
points  of  an  electric  circuit  to  the  actual  current  flowing  in  that 
portion  of  the  circuit.  Thus  considered,  no  assumptions  what- 
ever are  made  respecting  the  constancy  or  nature  of  the  quantity 
Ri.  When  this  quantity  is  obtained  by  a  measurement  of  the 

J? 

ratio  -j  ,  it  is  customary  to  call  it  an  ohmic  resistance.     Having 

been  thus  obtained  it  is  often  treated  as  if  it  might  be  used  as  in 
Eq.  (1),  and  is  often  incorrectly  considered  a  constant,  which  it 
is  not  unless  specifications  of  the  conditions  under  which  it  may  be 
so  considered  are  specifically  stated. 

As  a  matter  of  fact,  in  many  cases  in  which  resistances  are 
measured,  when  the  current  is  made  appreciable  the  conductor 

W 

heats,  and  the  ratio  -j  changes.     It  is  a  change  which  is  independ- 

ent of  external  temperatures,  which  are  always  assumed  to  be 
specified  when  a  resistance  is  designated  or  measured.  This  change 
of  resistance  with  the  current  used  in  making  the  measurement  is 
strikingly  illustrated  in  measuring  the  resistance  of  carbon  lamp- 
filaments  and  tungsten  lamp-filaments.  In  some  measurements 

7? 

made  by  the  author  the  ratio  y  for  a  carbon  lamp-filament  when 

cold  (that  is,  with  a  current  too  small  to  heat  it  appreciably)  was 

20  fi  53  5 

—  r^  =  127.1,  and  for  the  same  filament,  when  hot,  was 


105.9.  A  tungsten  lamp-filament  in  series  with  the  carbon  lamp- 
filament  gave,  cold, 

10.9  __ 

,       0162  -  67'28' 
and  hot, 

72'° 

0505 

W 

In  the  above  cases  the  ratio  y,  or  what  would  be  termed  the 

resistance,  is  very  dependent  upon  the  current  used  in  making  the 
measurements;  in  the  one  case  diminishing  and  in  the  other  case 
increasing  with  increase  of  current.  If  the  measuring  current 

f1 
were  continually  diminished,  the  ratio  y  would  approach  a  value, 

both  in  the  case  of  carbon  and  tungsten  filaments,  which  would  be 


18  MEASURING  ELECTRICAL  RESISTANCE  J       [ART.  105 

constant  for  any  particular  external  temperature.  This  constant 
value  may  be  called  the  true  ohmic  resistance  of  the  conductor 
at  the  external  temperature  at  which  the  measurement  is  made. 
As  the  value  of  every  resistance  varies  to  some  extent,  and  often 
very  greatly,  both  with  the  external  temperature  and  the  measur- 
ing current  used  to  determine  it,  it  is  seen  that  a  resistance  is  only 
completely  specified  when  both  of  these  conditions  are  fully 
specified. 

All  conductors,  used  for  conveying  current,  vary  more  or  less 
as  above  and  there  is  a  very  large  class  of  resistance  measurements 
where  the  quantity  measured,  and  called  a  constant  resistance, 
cannot  be  considered  a  constant  quantity  even  approximately 
unless  accompanied  by  a  precise  statement  of  the  conditions  under 
which  the  measurerrient  is  made.  Most  of  the  methods  usually 
described  for  the  measurement  of  resistance  seem  to  assume  this 
quantity  as  practically  constant  when  only  the  temperature  of  its 
surroundings  is  maintained  constant,  and  the  custom  is  to  treat 
the  quantity  from  the  viewpoint  of  Eq.  (1)  above.  In  many 
cases  this  procedure  is  justified,  for  it  is  true  that  by  using  special 
alloys  and  small  measuring  currents  many  kinds  of  resistance- 
determinations  are  made  where,  to  a  first  approximation  at  least, 
the  resistance  is  a  constant  quantity  which  may  be  multiplied 
into  a  current  of  any  reasonable  value  and  so  give  the  potential- 
drop  at  its  terminals. 

Broadly  speaking,  however,  metallic  conductors  should  be 
divided  into  two  classes  —  conductors  intended  to  serve  as 
definite  resistance  (as  those  used  in  resistance-standards,  resist- 
ance-measuring instruments,  rheostats,  and  wherever  the  flow  of 
current  is  to  be  restrained)  and  conductors  intended  to  convey 
electric  power  with  as  small  a  loss  of  power  as  possible.  The  former 
class  of  conductors  is  given  intentionally  a  high  resistivity  and  con- 
sists of  alloys  which  change  little  in  resistivity  with  the  temper- 
ature-rise produced  by  the  current  or  with  temperature-changes 
in  the  surrounding  medium.  The  latter  class  of  conductors  con- 
sists chiefly  of  the  pure  metals,  copper  and  aluminum,  and  is  given 
as  low  a  resistivity  as  possible.  In  these  conductors  a  large  tem- 
perature coefficient  cannot  be  avoided  and  they  change  greatly, 
about  0.4  of  1  per  cent  per  degree  C.,  with  change  of  temperature 
from  any  cause.  In  measuring  conductors  of  the  first  class,  we 
may  conveniently  consider  resistance  as  a  constant  property  of  a 


ART.  105]      EXTENT  OF  ELECTRICAL  MEASUREMENT  19 

circuit  at  a  given  external  temperature  which,  when  once  deter- 
mined, will  give  the  potential-drop  if  multiplied  by  the  current,  it 
being  merely  assumed  that  the  current  is  kept  within  reasonable 
limits.  In  measuring  conductors  of  the  second  class,  it  is  more 
convenient  to  regard  resistance  as  a  quantity  which  merely 

P1 

expresses  the  ratio  y  •    If  the  resistance  of  such  a  conductor  is 

determined  in  terms  of  the  resistance  of  the  conductor  of  the  first 
class,  the  value  obtained  will  be  indefinite  unless  very  complete 
statements  are  furnished  respecting  the  external  temperature  and 
the  current  used  in  making  the  measurement.  The  difficulty  of 
doing  this  has  made  it  customary  to  determine,  not  the  resistance 
of  such  conductors,  but  their  conductivity  in  terms  of  the  conduc- 
tivity of  a  standard  conductor  of  the  same  kind.  By  the  methods 
employed  in  conductivity-measurements,  both  the  temperature  of 
the  surroundings  and  the  heating  effect  of  the  measuring  current 
are  caused  to  act  alike  upon  the  standard  and  the  sample,  and 
hence  do  not  require  special  consideration  or  specification. 

The  standard  methods  employed  for  measuring  resistance  and 
conductivity  differ  considerably,  and  whether  it  is  better  to  make 
a  resistivity  or  a  conductivity  determination  will  usually  depend 
upon  the  class  of  conductors  under  consideration.  We  shall1  con- 
sider first  those  methods  of  measuring  resistance  where,  to  a  first 
approximation,  resistance  may  be  treated  as  a  quantity  which  is 
constant  under  ordinary  conditions,  leaving  to  a  later  section  a 
consideration  of  the  methods  employed  in  the  measurement  of 
conductors  of  the  second  class. 

Resistances  may  be  subdivided  conveniently  into  medium,  high, 
and  low  resistances,  as  the  same  methods  of  measurement  are  not 
equally  well  adapted  to  all  three. 

The  electrical  instrument  which  is  most  universally  available 
is  a  voltmeter.  These  instruments  are  supplied  with  a  fixed 
internal  resistance  the  value  of  which  is  usually  stated  upon  the 
instrument,  or,  if  not  stated,  may  be  easily  determined.  As 
many  kinds  of  resistance  measurements  may  be  made  with  this 
instrument  alone,  those  methods  which  are  useful  will  now  be 
considered. 


CHAPTER  II. 


RESISTANCE  MEASURED  WITH  DEFLECTION  INSTRU- 
MENTS; VOLTMETER  AND  AMMETER  METHODS. 

200.  Assumptions.  —  The    methods    generally    involve    the 
assumptions:    (a)  that  the  scale  divisions  of  the  deflection  instru- 
ment are  so  laid  off  that  the  readings  of  the  instrument  are  pro- 
portional to  the  current  passing  thru  it,  and  (b)  that  the  E.M.F., 
or  the  source  of  current,  remains  constant  while  taking  successive 
readings,  and  (c)  that  the  internal  resistance  of  the  source  of  cur- 
rent is  negligibly  small  compared  with  the  other  resistances  in  the 
circuit. 

The  instrument  generally  used  is  a  voltmeter,  but  it  may  be 
a  millivoltmeter  with  a  known  resistance  connected  in  series  with 
it,  or  a  galvanometer  which  has  a  proportional  scale  and  a  known 
resistance  R. 

201.  Voltmeter  Method.     Circuit  Includes  a  Known  Resist- 
ance.    Method  I.  —  To  meet  the  general  case  we  shall  suppose 

r  r 

r • WWVAAA 


1  l-vi-J 


R 
11 


FIG.  201. 


that  there  is  a  resistance  r  in  the  circuit  as  indicated  in  Fig.  201. 
With  the  connections  as  shown  in  I  the  current  thru  the  deflec- 
tion instrument  will  be 


a) 


where  x  is  the  resistance  to  be  measured, 

R  the  resistance  of  the  instrument, 
and       V  the  E.M.F.  of  the  source. 

20 


ART.  201]        VOLTMETER  AND  AMMETER   METHODS  21 


Also,  C  =       '  (2) 

where  V\  is  the  E.M.F.  at  the  terminals   of  the   instrument. 
From  Eqs.  (1)  and  (2) 

7ifl  +  Fir  +  VlX  m 

~R~ 

With  the  connections  as  shown  in  II  the  current  thru  the  deflec- 

tion instrument  will  be 

V          Vn 
Cr  =  =  —  >  (4) 

where   Vz  is  the  E.M.F.  at  the  terminals  of  the  instrument, 
whence, 

F  =  TV  +  ™. 

ri 

From  Eqs.  (3)  and  (5)  we  derive, 


_ 


V, 

Since  it  is  assumed  that  the  deflections  of  the  instrument  used 
are  proportional  to  the  E.M.F.'s  at  its  terminals,  we  can  write 
Vi  =  kdij  Vz  =  kd2,  where  k  is  a  constant  and  di  and  dz  are  the 
deflections  corresponding  to  the  voltages  V\  and  V*.  Hence 
Eq.  (6)  may  be  written 


Equation  (7)  shows  first,  that,  if  the  law  of  the  scale  of  the 
instrument  is  one  of  proportionality,  it  is  not  necessary  that  the 
deflections  should  indicate  volts,  millivolts  or  any  particular 
quantity.  It  follows  that  for  measuring  resistances  in  this  way 
one  may  use  a  voltmeter,  a  millivoltmeter  with  some  extra  known 
resistance  in  series  with  it,  or  a  galvanometer.  If  a  150-  volt 
voltmeter  be  used,  then  the  current  may  be  obtained  from  a  110- 
volt  D.C.  circuit,  but  if  a  millivoltmeter  or  a  galvanometer  be 
used,  one  or  more  cells  of  storage  battery  will  suffice.  In  this 
method,  as  generally  applied,  the  resistance  r  is  zero,  in  which  case 

x  =  d-R,  (8) 


The  following  actual  sets  of  readings,  taken  by  careful  but 
untrained  observers,  will  serve  to  illustrate  the  method  above,  as 


22 


MEASURING  ELECTRICAL  RESISTANCE         [ART.  202 


expressed  in  Eq.  (7),  and  to  indicate  the  precision  which  may  be  ex- 
pected under  the  arranged  conditions.  The  deflection  instrument 
was  a  Weston  millivoltmeter,  temporarily  supplied  with  a  series 
resistance  obtained  from  a  resistance  box,  such  that  R  =  102.3 
ohms  (Fig.  201).  The  source  of  current  was  one  cell  of  storage 
battery. 

The  resistances  measured  were  accurate  coils  in  a  resistance  box. 
r  was  taken  100  ohms. 

The  table  below  gives  the  results  obtained. 


*< 

d, 

x,  given 

x,  observed 

Per  cent  error 

50.1 

50.2 

1 

0.4 

-60.0 

47.9 

50.2 

10 

9.7 

-30.0 

45.5 

50.2 

21 

20.8 

-  0.95 

42.1 

50.2 

39 

38.9 

-  0.25 

35.7 

50.2 

82 

82.2 

+  0.25 

33.5 

50.2 

101 

100.8 

-  0.19 

28.3 

50.2 

157 

156.6 

-  0.25 

19.5 

50.2 

321 

318.5 

-  0.77 

12.1 

50.2 

638 

636.7 

-  0.17 

6.6 

50.2 

1,283 

1336.0 

+  4.1 

3.6 

50.2 

2,559 

2731.0 

+  6.7 

1.9 

50.2 

5,117 

5143.0 

+  0.55 

1.0 

50.2 

10,241 

9953.0 

-  2.8 

It  should  be  noted  that  the  error  increases  as  the  difference, 
di  —  di,  becomes  smaller,  and  that  a  precision  of  better  than  one 
per  cent  may  hardly  be  expected  in  the  range  20  ohms  to  5000 
ohms.  It  may  happen,  however,  that  the  only  E.M.F.  avail- 
able is  such  as  to  deflect  the  instrument  off  its  scale  when 
connected  directly  to  it  and  that  it  is  necessary  to  include  a  re- 
sistance, as  r  or  x,  Fig.  201,  in  the  circuit  at  all  times.  If  this 
resistance  r,  as  well  as  the  resistance  x  is  unknown,  the  values  of 
r  and  x  may  both  be  obtained. 

The  following  method  may  be  easily  applied,  tho  the  formula 
to  express  the  result  is  rather  lengthy. 

202.  Voltmeter  Method.  Circuit  Includes  an  Unknown  Re- 
sistance. —  Referring  to  Fig.  201,  interchange  the  resistances 
r  and  x,  and  repeat  the  measurements  as  made  in  case  (1).  The 
deflection  obtained  when  the  circuits  are  arranged  as  in  I  will  be 
the  same  as  before,  hence  d\  remains  di,  while  the  deflection  ob- 
tained when  the  circuits  are  arranged  as  in  II,  but  with  r  and  x 


ART.  2031       VOLTMETER  AND  AMMETER  METHODS 


23 


interchanged,  will  be  d%;  then,  we  have,  in  the  same  manner  that 
Eq.  (7)  (§  201)  was  obtained, 

_  W-di)  (R+x) 


The  resistance  r  may  now  be  eliminated  from  Eq.  (7)  (§  201)  and 
Eq.  (1)  (§  202),  and  we  obtain 


,,- 

-  d,'  (dz  -  dj' 


or,  eliminating  x, 

Rhfa'-dd 
dM  -  d2  (dzf  -  dd  ' 

203.  Voltmeter  Method.  Circuit  Includes  a  Known  Resist- 
ance. Method  II.  —  The  usual  method  of  making  this  measure- 
ment is  to  connect  the  known  resistance  r  in  series  with  the 
unknown  resistance  x  and  read  the  deflections  obtained  when  first 
the  unknown  and  second  the  known  resistance  are  shunted  with 
the  deflection  instrument.  Make  the  connections  as  indicated  in 
I  and  II,  Fig.  203. 


0 

1 

T 

X 

r 

L 


vwvwvwv 


Reads  d\ 


Reads  cfe 


II 


FIG.  203. 


If  the  assumption  is  made  that  the  resistance  R  of  the  deflec- 
tion instrument  is  500  or  more  times  that  of  either  the  resistance 
r  or  the  resistance  x,  then  we  have  simply 


7    ?       '-'•*•       *  —  ~T~ 

r      d2  d2 


(1) 


where  di  is  the  deflection  with  the  connection  I  and  d2  the  deflec- 
tion with  the  connection  II. 

If  one  wishes,  however,  to  measure  the  resistance  of  an  incan- 
descent lamp  in  this  way,  using  a  voltmeter,  the  above  assump- 
tion would  not  be  permissible  for  precise  work,  as  the  voltmeter 
would  shunt  from  the  resistance,  at  the  terminals  of  which  it  is 


24 


MEASURING  ELECTRICAL  RESISTANCE         [ART.  204 


connected,  a  portion  of  the  current  which  may  not  be  neglected. 
If  the  resistance  R  of  the  deflection  instrument  is  known,  and 
if  the  assumption  be  made  that  the  main  current  is  kept  un- 
changed (as  by  means  of  a  large  resistance  Q)  when  the  voltmeter 
is  connected,  as  in  I  and  in  II  (Fig.  203),  the  value  of  x  may  be 
precisely  expressed  by  the  relation 


_ 
~ 


(  } 


Eq.  (2)  is  easily  deduced.     If  we  put  this  expression  in  the  form 

dp 


x  = 


r  (d,  - 
R 


it  is  seen  that,  when  R  is  very  great  as  compared  with  r,  the 
expression  takes  the  same  form  as  Eq.  (1).  If  r  nearly  equals  x, 
the  deflection  d\  will  be  nearly  the  same  as  the  deflection  d2  and 
Eq.  (1)  again  becomes  applicable. 

The  above  method,  as  expressed  in  Eq.  2,  was  applied  to  the 
measurement  of  the  resistance  of  a  60- volt  tungsten  lamp  operated 
first  on  13  and  then  on  40  volts.  A  Weston  voltmeter  reading 
to  150  volts  was  used.  The  resistance  R  of  the  voltmeter  was 
15,660  ohms.  The  resistance  r  in  series  with  the  lamp  was  260 
ohms.  The  results  obtained  are  exhibited  in  the  table  below: 


di 

d2 

X 

Remarks 

13.1 

47.3 

71.2 

The  resistance  of  the  tungsten 

33.9 

83.3 

104.8 

lamp  increases  1.61  ohms  per 

volt  in  range  13  to  40  volts. 

It  should  be  noted  that,  in  order  to  apply  this  method  of  measur- 
ing the  resistance  of  a  lamp  while  burning,  a  known  resistance  r 
must  be  in  series  with  the  lamp,  which  will  carry  the  same  current 
as  the  lamp.  It  is  also  necessary  to  have  for  the  measurement  a 
voltage  higher  than  the  voltage  which  is  to  be  applied  at  the  lamp- 
terminals  when  the  lamp  is  in  service.  Compare  this  measure- 
ment with  another  measurement  made  upon  the  same  lamp  by 
the  method,  par.  209  (Fig.  209). 

204.  Comparing  Potential  Drops  with  a  Deflection  Instru- 
ment; Special  Case.  —  There  is  a  special  case  in  which,  when 
two  resistances  are  joined  in  series,  the  ratio  of  the  two  deflections 


ART.  204]        VOLTMETER  AND  AMMETER  METHODS 


25 


obtained  (when  the  terminals  of  the  instrument  are  first  con- 
nected to  the  terminals  of  the  one  resistance  and  then  to  the 
terminals  of  the  other  resistance)  gives  the  ratio  of  the  resistances, 
whatever  be  the  resistances  themselves  or  the  resistance  of  the 
deflection  instrument. 


M.V. 


t — ^A/www^AAAA/y — wwwwvwww-" 
r  b  x- 


,Ba 


FIG.  204. 

The  necessary  conditions  which  must  be  assumed  to  hold,  to 
make  the  above  statement  rigidly  true  are:  that  (Fig.  204)  the 
resistance  of  the  battery  Ba  together  with  the  leads  from  the 
battery  to  the  points  a  and  c  is  zero,  and  that  the  instrument 
M.V.  may  be  joined  to  a  point  b  which  is  a  potential  terminal 
common  to  both  r  and  x.  Both  of  these  conditions  can  be  often 
filled  in  practice  very  approximately. 

Calling  di  the  deflection  of  the  deflection  instrument  M.V. 
when  joined  to  b  and  c,  and  c?2  the  deflection  when  joined  to  a  and 
6,  and  g  the  resistance  of  the  deflection  instrument,  we  have  the 
following  proof  of  the  above  statement.  When  the  deflection 
instrument  is  joined  to  b  and  c,  the  current  Ci  thru  the  instru- 
ment is 

V          _x  ~  Vx 

(J 


or 


r-h 


gx 


g  (r  +  x)  +rx 


(1) 


Similarly,  when  the  instrument  is  joined  to  the  terminals  a  and 
b,  the  current  C2  thru  the  instrument  is 

C   =  Vr  (2) 

g  (r  +  x)  +  rx 

As  the  denominators  of  Eqs.  (1)  and  (2)  are  the  same,  we  have 


and,  if  the  instrument  deflections  are  proportional  to  the  current 
thru  it, 


26 
hence, 


MEASURING   ELECTRICAL  RESISTANCE         [ ART.  205 


(3) 


If  there  is  resistance  in  the  battery  or  the  battery  leads,  relation 
(3)  is  not  rigidly  true,  nor  is  it  rigidly  true  if  r  and  x  are  each 
provided  with  potential  terminals  so  that  the  point  b  cannot  be 
made  common  to  both  resistances.  In  cases,  however,  where  the 
two  resistances  are  considerably  lower  than  the  resistance  of  the 
deflection  instrument  the  error  resulting  by  applying  Eq.  (3)  is 
too  small  to  deserve  consideration.  Relation  (3)  would  also  be 
rigidly  true  under  all  circumstances  in  the  single  case  when  r  and 
x  are  equal. 

205.  Voltmeter  Method  Using  a  Shunt.  —  It  is  sometimes  de- 
sirable to  shunt  the  deflection  instrument  with  a  known  resistance 
S.  In  this  case  the  unknown  resistance  may  be  determined  in  a 
manner  similar  to  that  given  under  method  I,  Fig.  201. 

7?  Sf 
Thus,  referring  to  Fig.  205,  let  P  =  ^-  — »  be  the  resistance  of 

£1  -}-  O 


T 

1 
-       1 

'  T® 

J 

•Y 

1 
1 
4 

JJ 

FIG.  205. 

the  instrument  when  shunted.     With  p  joined  to  a,  the  current  in 
the  line  will  be 

r         V         V1 

-jr+n'T' 

where  V  is  the  potential  of  the  source  and  Vi  the  potential  at  the 
terminals  of  the  instrument. 

If  the  deflections  of  the  instrument  unshunted  are  proportional 
to  the  E.M.F.  at  its  terminals,  they  will  also  be  proportional  when 
the  instrument  is  shunted.  Therefore,  we  can  write  V  —  kD,  and 
Vi  =  kd.  Hence, 

kD        kd  P(D-d] 


=     '    or  x= 


d 


ART.  2061       VOLTMETER  AND  AMMETER  METHODS 


27 


RS     D-d 


or 


(2) 


R+S      d 
The  deflection  D  will  be  obtained  by  joining  p  to  6. 

The  method  as  expressed  in  Eq.  (2)  is  illustrated  by  the  follow- 
ing measurements: 

A  millivoltmeter  was  used  with  resistance  in  series  with  it  so 
that  R  =  100.3  ohms.  The  resistance  R  was  shunted  with  a 
resistance  S  =  10.0  ohms,  so  that 

D      10  X  100.3      n  nno    , 
P-  10  +  100.3  "g-Orcohms. 

One  cell  of  storage  battery  was  used  for  the  source  of  current  and 
the  resistances  measured  were  coils  in  a  resistance  box.  The 
connections  were  made  as  in  Fig.  205,  and  the  results  obtained 
are  those  given  in  the  table  below,  calculated  by  the  formula 

D-d 


x  =  9.093 


d 


D 

d 

X 

true  value 

folia 

Per  cent  error 

100.1 

90.3 

1 

0.987 

-1.3 

100.1 

64.9 

5 

4.93 

-1.4 

100.1 

31.6 

20 

19.71 

-1.5 

100.1 

15.6 

50 

49.25 

-1.5 

100.1 

8.4 

100' 

99.24 

-0.76 

100.1 

0.9 

1000 

1003.00 

+0.3 

206.  Deflection  Method.  Resistance  Measured  by  Substitu- 
tion. —  This  is,  perhaps,  the  simplest  method  employed  for  meas- 
uring a  resistance,  and  is  useful  in  certain  cases,  especially  where 
insulation  measurements  are  to  be  made. 

For  the  sake  of  generality,  suppose  the  resistance  R  of  the  de- 
flection instrument  is  shunted  with  a  resistance  S. 
RS 


LetP  = 


be  the  shunted  value  of  the  resistance  of  the 


R  +  S 
instrument. 

Let  r  be  the  resistance  of  the  source  of  current  and  Q  the  known 
resistance.  Then  (Fig.  206a)  if  V  is  the  E.M.F.  of  the  source  and 
C  the  current  flowing,  we  have,  with  p  joined  to  a, 

V 


C  = 


=  kD 


r+x 
where  k  is  a  constant  and  D  is  the  deflection  obtained. 


28 


MEASURING  ELECTRICAL  RESISTANCE         [ART.  206 


With  p  joined  to  b  the  current  C'  flowing  will  be 

F 


rif  _ 

r  +  Q  +  P 
where  d  is  the  deflection  obtained. 


kd, 


(2) 


FIG.  206a. 


From  Eqs.  (1)  and  (2) 
d 


(3) 


The  value  of  x  cannot  be  obtained  from  Eq.  (3)  unless  the  re- 
sistance r  of  the  source  of  current  is  either  known  or  is  negligible. 
This  method  is  scarcely  ever  employed  except  in  those  cases  where 
r  may  be  neglected.  With  this  assumed, 

In  the  practical  application  of  this  method  to  the  measurement 
of  high  resistances  the  shunted  resistance  P  of  the  deflection  in- 
strument may  also  be  neglected,  in  which  case 

*  =  ^Q.  (5) 

The  application  of  this  method,  and  its  modifications,  to  the 
measurement  of  the  insulation  resistance  of  cables  is  better  re- 
served for  a  later  chapter. 

If  the  resistance  Q  is  a  rheostat  which  can  be  varied  in  small 
steps,  then  Eq.  (4)  may  be  greatly  simplified  and  the  measure- 
ment made  with  greater  accuracy,  if  we  so  adjust  Q  that  the  de- 
flection d  is  one  half  the  deflection  D.  In  this  case 

Q-P 


X  = 


(6) 


ART.  206]        VOLTMETER  AND  AMMETER   METHODS  29 

Or,  if  P  is  very  small  as  compared  with  x  or  Q,  we  have 

*-§,  '  (7) 

which,  indeed,  is  obvious. 

Again,  if  Q  is  varied  until  the  deflection  with  p  joined  to  Q  is 
the  same  as  the  deflection  with  p  joined  to  a,  then  x  —  Q.  In 
making  the  measurement  in  this  manner  we  substitute  for  the 
unknown  quantity  another  quantity  which  is  known  and  can  be 
varied  until  the  indicating  instrument  gives  exactly  the  same  in- 
dication whether  the  known  or  the  unknown  quantity  is  in  circuit. 

In  this  application  of  the  substitution  method  no  assumptions 
as  to  the  proportionality,  etc.,  of  the  indicating  instrument  are 
needed.  Only  two  requirements  need  to  be  met  for  obtaining 
very  high  precision;  first,  that  the  indicating  instrument  shall  be 
sensitive  to  small  changes  in  the  known  quantity  substituted,  and 
second,  that  this  quantity  may  be  varied  by  very  small  steps. 

The  following  method  of  measuring  the  resistance  of  an  elec- 
trolyte is  a  good  example  of  "the  substitution  method."  This 
method  was  described  by  Prof.  C.  F.  Burgess  in  Vol.  XI,  1907, 
page  225,  Trans,  of  the  Electrochem.  Soc.  The  method  of  Profes- 
sor Burgess  as  given  here  has  been  somewhat  modified  in  accord- 
ance with  a  suggestion  made  by  Dr.  Carl  Hering. 

In  Fig.  206b  is  shown  a  glass  tube,  of  known  cross-section  and 
with  a  graduated  scale,  immersed  in  an  electrolyte,  the  specific 
resistance  of  which  is  to  be  measured.  At  the  bottom  of  the  glass 
tube  is  a  fixed  electrode,  and  the  other  electrode  is  fastened  to  a 
rod  by  which  it  may  be  set  at  various  points,  as  p\  and  p%,  in  the 
glass  tube.  F  is  a  millivoltmeter  and  R  is  a  resistance  box  ad- 
justable in  small  steps.  The  millivoltmeter,  resistance  box,  and  the 
electrolyte  in  the  glass  tube  are  placed  in  circuit  with  a  constant- 
potential  storage  battery  Ba.  R  is  then  adjusted,  with  the  elec- 
trode S  set  at  pi  until  the  millivoltmeter  reads  near  the  top  of  its 
scale.  The  electrode  S  is  then  moved  up  to  p2  and  resistance  is 
plugged  out  of  the  box,  until  the  millivoltmeter  reads  the  same  as 
before.  Then,  without  regard  to  the  polarization  of  the  electrodes, 
the  resistance  Rt  plugged  out  of  the  box  is  equal  to  the  resistance 
of  the  electrolyte  between  the  points  p\  and  p2- 

If  s  is  the  cross-section  of  the  tube,  and  I  is  the  length  p\  to  pz 
and  Rt  is  the  resistance  plugged  out,  the  specific  resistance  of  the 


30 


MEASURING  ELECTRICAL  RESISTANCE         [ART.  207 


electrolyte,  at  the  temperature  t  at  which  the  measurement  is 
made,  is 

sRt 


Pt  = 


I 


(8) 


Compare  this  method  with  the  method  of  Kohlrausch  given  in 
par.  1120. 


FIG.  206b. 

207.  Voltmeter  Method.  Circuit  Forms  Loop  of  Three  Un- 
known Resistances,  Two  of  which  are  to  be  Determined.  —  The 

following  method  is  a  more  general  case  of  the  measurement  of 
resistance  with  a  deflection  instrument  and  is  especially  useful  for 
determining  the  insulation  resistance  of  a  two-wire  interior  wiring 
system,  while  the  power  is  on.  The  method  was  first  described 
by  the  author  in  the  Electrical  World  and  Engineer,  pages  966-967, 
May  21,  1904. 

The  theory  only  of  the  method  is  given  here,  its  application  to 
insulation  measurements  of  live  wiring  systems  being  left  to  a 
later  chapter.  Reference  should  be  made  to  Fig.  207,  I,  II,  and 
III,  for  the  manner  of  making  the  connections  and  the  meaning 
of  the  symbols  used. 


ART.  207]       VOLTMETER  AND  AMMETER  METHODS 

C XTX  Reads  D 

R 


31 


( vwwvwvw 


B3 


y 
^wwwvww 


B2 


c; 


• — => — wwwwvv 


AWVWVWVW 


w 

II 


II- 


•A/WWWW *— tB2 


Reads  C? 


VWWWW\/ — i 


III 

FIG.  207. 

Cij  C2,  etc.,  are  currents  and  R  is  the  known  resistance  of  the 
deflection  instrument  used.  D,  di,  and  d2  are  deflections  each  of 
which  is  assumed  in  all  cases  to  be  proportional  to  the  E.M.F. 
applied  at  the  terminals  of  the  instrument.  Measurements  of  this 
kind  would  ordinarily  be  made  with  a  Weston  voltmeter,  reading 
to  150  volts,  and  110  volts  on  the  supply  mains,  or  with  a  milli- 
voltmeter,  having  a  known  resistance  of  from  200  to  500  ohms  in 


32  MEASURING  ELECTRICAL  RESISTANCE         [ART.  207 

series  with  it,  and  current  supplied  from  one  or  two  cells  of  storage 
battery. 

If  direct  current  is  supplied  to  the  system,  a  direct-current 
deflection  instrument  must  be  used,  as  a  Weston  voltmeter,  a 
millivoltmeter  with  a  series  resistance,  or  a  galvanometer  of  known 
resistance. 

If  the  current  is  alternating,  then  an  alternating-current  Weston 
voltmeter  may  be  used,  or  other  alternating-current  deflection 
instrument  of  the  electrodynamometer-type  with  a  scale  marked 
to  give  indications  which  are  directly  proportional  to  the  E.M.F. 
at  the  instrument  terminals  or  to  the  current  thru  it. 

The  resistances  x\  and  Xz  are  determined  by  knowing  R,  the 
resistance  of  the  deflection  instrument,  and  by  taking  three  instru- 
ment readings: 

1st.  Determine  the  deflection  when  the  instrument  is  connected 
to  Bl  and  Bz  (Fig.  207,  I).  Call  this  D, 

2nd.  Determine  the  deflection  when  the  instrument  is  connected 
to  Bi  and  B3  (Fig.  207,  II).  Call  this  di. 

3rd.  Determine  the  deflection  when  the  instrument  is  connected 
to  B3  and  Bz  (Fig.  207,  III).  Call  this  d*. 

If  in  case  1  the  deflection  goes  off  the  scale  or  in  cases  2  and  3 
the  readings  are  but  a  very  small  fraction  of  the  total  scale,  the 
method  is  not  applicable,  without  modification,  in  the  former  case, 
and  in  the  latter  case  the  resistances  Xi  and  Xz  are  too  high  to  be 
satisfactorily  measured  by  this  method.  Having  taken  the  above 
three  readings  it  will  be  shown  that 

R  (D  —  di  —  dz)  ,t  x 

Xi  =  -        —j »  (1) 

dz 

and 

R  (D  —  di  —  dz)  ,0\ 

Xz  =  -j -    •  (A) 

As  the  resistance  y  does  not  enter  into  Eqs.  (1)  and  (2)  it  may 
have  any  value  without  influencing  the  result. 

It  is  to  be  noted  that,  as  x\,  Xz,  and  y  are  in  series  and  form  a 
closed  loop,  this  method  determines  the  resistances  of  portions  of 
a  closed  circuit  without  cutting  it. 

We  further  note  from  Eqs.  (1)  and  (2)  that 


ART.  2071        VOLTMETER  AND  AMMETER  METHODS  33 

and  that  if  Xi  =  infinity,  the  deflection  d2  =  0,  and  hence, 


This  is  the  ordinary  expression  used  in  measuring  a  resistance  with 
a  voltmeter,  and  has  the  same  form  as  Eq.  8,  par.  201. 

Equations  (1)  and  (2)  above  are  obtained  as  follows:  Let  k  be 
the  constant  of  the  deflection  instrument,  such  that  the  E.M.F.'s 
at  the  instrument  terminals  are  proportional  to  the  deflections. 
Then, 


Ci- ^— ,  (5) 

/I#i 
*^2    ' /? 

c°=— *TST--  ^ 

#1  +: — r~H 


^  /  _  X\          „     _  KCli 

Cl  :=^~^Cl~  IT 


or.  d=       ^^  (7) 

kdt 

R 


Cr  _         •"2        f 
2    —   D     i     „    ^2  —   ~B~» 


C2  =  .  (8) 

(9) 


Hence  we  have  the  two  relations, 

kD  kdi(R+xi) 


Rx\ 


A;Z)             /bc?2(jR+^2)  /ir,x 

and  5-—  =  — ^ '  (10) 


The  factor  k  cancels  from  equations  (9)  and  (10)  and  xl  and  £2  are 
found,  by  simple  algebraic  transformations,  to  have  the  values 
given  above  in  Eq.  (1)  and  Eq.  (2).  If  any  one  of  the  equations 
(1),  (2)  and  (4)  are  solved  for  R,  it  is  seen  that  we  have  a  method, 
when  one  resistance  is  known,  of  determining  the  internal  resist- 
ance of  a  deflection  instrument.  The  method  is  often  useful  for 
finding  the  resistance  of  a  galvanometer  when  no  other  instrument 
is  available.  As  the  E.M.F.  of  a  single  cell  of  battery  will  throw 


34 


MEASURING   ELECTRICAL  RESISTANCE         [ART.  208 


a  galvanometer  off  its  scale,  it  is  necessary  to  use  a  feeble  E.M.F. 
which  may  be  obtained  by  opposing  two  nearly  equal  cells,  or  by 
taking  the  drop  between  two  near  points  on  a  low-resistance  wire. 
The  above  method  is  illustrated  by  the  following  measurements : 
The  deflection  instrument  used  was  a  millivoltmeter  with  a  resist- 
ance in  series  with  it,  so  that  [Eqs.  (1)  and  (2)]  R  =  102.3  ohms. 
The  other  quantities  in  the  table  below  have  the  same  meanings 
as  in  Eqs.  (1),  (2)  and  in  Fig.  207.  The  source  of  current  was  a 
single  cell  of  storage  battery  and  the  resistances  measured  were 
manganin  resistance  coils.  For  the  sake  of  generality  a  resistance  y 
(Fig.  207)  of  200  ohms  was  used  to  shunt  the  resistances  Xi  and  x*. 
The  results  obtained  are  given  in  the  table  below. 


D 

di 

•d, 

X 

true  value 

X 

found 

Per  cent 
error 

100.5 

31  5 

63.0 

Zi  =    10 

z2=  20 

9.74 

19.48 

-2.6 
-2.6 

100.8 

72.2 

14.4 

zi  =  100 

x,=  20 

100.88 
20.12 

+0.88 
+0.6 

It  should  be  noted  that  in  the  first  case  the  difference 
D  —  (di  +  d2)  =  6  divisions,  only,  and  in  the  second  case  it 
equals  14.2  divisions  and  that  the  precision  is,  as  should  be  ex- 
pected, better  in  the  latter  case. 

The  theoretical  precision  which  can  be  obtained  by  this  method 
is  fully  discussed  in  par.  104. 

208.  Limitations  of  Voltmeter  Methods.  —  It  remains  to 
point  out  the  limitations  of  voltmeter  methods  when  applied  to 
the  measurement  of  resistances  of  over  a  megohm. 


-in, 


FIG.  208 

Let  m  and  mi  be  two  direct-current  mains  between  which  the 
potential  is  e. 

Let  V  be  the  voltmeter  of  internal  resistance  R  and  let  re  be  a 
resistance  (over  a  megohm)  which  is  to  be  measured.  With  the 


ART.  209]        VOLTMETER  AND  AMMETER  METHODS  35 

connections  made  as  in  Fig.  208,  first  put  the  switch  s  on  a  and 
read  the  deflection  of  the  voltmeter,  which  call  d2,  then  on  6  and 
read  the  deflection,  which  call  d\.  By  the  theory  and  formula 
given  in  par.  201,  Eq.  (8),  we  have 


Ordinary  conditions  would  be  met  by  assuming  that  the  mains 
furnish  110  volts,  that  the  voltmeter  has  one  division  to  the  volt 
and  reads  150  volts  at  the  limit  of  its  scale,  and  that  it  has  a  re- 
sistance of  100  ohms  to  the  volt  or  a  total  of  15,000  ohms.  We 
shall  then  have  d2=  110  and  R  =  15,000. 

Solving  Eq.  (1)  for  d\,  we  find 

Rd2 

"  x  +  R 

If  we  assume  that  x  —  106  ohms,  or  one  megohm,  we  find  d\  = 
1.625  +  divisions. 
The  expression  for  calculating  the  maximum  percentage  error 

is, 

[See  Eq.  (4),  §  104]  where  -  is  the  fractional  part  of  the  larger 

deflection  which  can  be  read.  Suppose  that  the  readings  are 
taken  with  great  care  so  that  they  are  accurate  to  within  0.05  of 
a  division,  then 


_ 

p      20X110"  2200' 

and  we  find  by  Eq.  (2)  that 

p  110          „  110 +  1. 625  v, 

Ep  ~  2200  X  1.625  X  110  -  1.625  * 

This  is  a  fairly  representative  case  and  it  shows  that  one  megohm 
is  about  the  largest  resistance  which  can  be  measured  with  a 
150-volt  voltmeter,  with  even  very  moderate  precision.  If  the 
voltmeter  is  a  300-volt  instrument  and  this  potential  is  available 
the  accuracy  would  of  course  be  correspondingly  increased. 

209.  Resistance  Measured  with  a  Voltmeter  and  an  Ammeter. 
—  Connect  the  ammeter  in  series  with  the  resistance  and  the  volt- 


36 


MEASURING  ELECTRICAL  RESISTANCE         [ART.  209 


meter  in  shunt,  as  indicated  in  Fig.  209.     Calling  R  the  resistance 
of  the  voltmeter,  x  the  resistance  to  be  measured,  /  the  current 


Voltmeter 


FIG.  209. 

measured  by  the  ammeter,  and  V  the  potential  at  the  terminals 
of  the  voltmeter,  we  have,  by  Ohm's  law, 
Rx        V 


x  = 


(1) 


It  appears  from  Eq.  (1)  that  if  R  is  very  large  as  compared  with 
y 

V  the  term  ~  can  be  neglected. 
K 

As  the  above  method  is  often  used  to  measure  the  resistance  of 
an  incandescent  lamp  while  carrying  full  load  current,  the  follow- 

y 

ing  example  is  given  to  show  the  importance  of  the  term  -^  in 

•  •tc 

this  case:  Let  the  current  /  be  measured  with  an  ammeter, 
which  has  a  full  scale  value  of  1  ampere,  and  let  the  voltage  drop 
over  the  lamp  be  read  with  a  150-volt  voltmeter  which  has  100 
ohms  resistance  to  the  volt  or  a  total  of  15,000  ohms.  If  the  line 
current  is  0.5  ampere,  then  the  true  resistance  of  the  lamp,  if  the 
voltmeter  reads  110  volts,  is, 

110 


x  = 


0.5- 


110 


=  223.1  ohms. 


15,000 

If  the  last  term  in  the  denominator  is  neglected  we  obtain  for  the 
value  of  the  resistance, 


x'  =  ^r  =  220  ohms. 

This  result  is  1.39  per  cent  lower  than  the  true  value,  which  is 
223.1  ohms. 


ART.  210]       VOLTMETER  AND  AMMETER  METHODS 


37 


The  practical  way  to  measure  the  resistance  of  a  110- volt  in- 
candescent lamp  by  this  method,  using  Weston  instruments,  would 
be  to  divide  the  volts  read  by  the  current  and  add  1.4  per  cent 
to  the  result. 

A  trial  was  made  of  this  method  using  the  same  tungsten  lamp 
and  voltmeter  employed  in  the  test  to  illustrate  the  method 
shown  in  Fig.  203.  The  resistance  of  a  carbon  60- volt  lamp  joined 
in  series  with  the  tungsten  lamp  was  determined  at  the  same 
time.  The  resistance  R  of  the  Weston  voltmeter  was  15,660 
ohms.  The  ammeter  was  a  Weston  instrument  reading  1  ampere 
fora  full  scale.  The  results  were  calculated  by  Eq.  (1),  par.  209, 
and  are  exhibited  in  the  table  below: 


/ 

V 

X 

Remarks 

Tungsten  Lamp. 

0.479 

65.8 

138.5 

The  resistance  of  the  tungsten 

0.317 

32.7 

103.8 

lamp  increases  1.05  ohms  per 

volt  in  range  33  to  66  volts. 

Carbon  Lamp. 

0.620 

65.7 

106.6 

The    resistance    of    the    carbon 

0.270 

32.8 

122.3 

lamp  decreases  0.477  ohm  per 

volt  in  range  33  to  66  volts. 

Calculating  the  resistance  of  the  tungsten  lamp  at  33.9  volts 
from  the  resistance  found  at  32.8  volts  on  the  assumption  that  the 
resistance  increases  1.05  ohms  per  volt,  we  find  this  resistance  to 
be  105.06  ohms.  By  the  method  of  par.  203,  we  found  the  resist- 
ance at  this  voltage  to  be  104.8  ohms.  The  two  measurements 
are  thus  seen  to  be  in  agreement  within  0.25  of  1  per  cent. 

210.  Remarks  Upon  the  Methods  of  Chapter  II.  —  The 
methods  given  above  for  measuring  resistance  with  deflection 
instruments  of  convenient  type  are  the  ones  which  are  most  often 
used  and  are  best  suited  to  practice.  They  may  be  modified  and 
extended  in  a  variety  of  ways,  but  the  fundamental  principles 
involved  would  not  be  altered.  The  methods  as  given  are 
typical  and  will  cover  almost  every  case  that  may  arise.  They 
are  sufficient  to  illustrate  the  principles  involved,  and,  if  these  are 
understood,  modifications  may  easily  be  made  to  meet  any  special 
requirements.  In  considering  the  theoretical  precision  of  the 
deflection  methods  given  above,  reference  should  be  made  to  the 
elementary  discussion  of  the  theory  of  errors  in  Chapter  I. 


38  MEASURING  ELECTRICAL  RESISTANCE         [ART.  211 

2ii.  Ohmmeters  and  Meggers.  —  These  instruments  are 
devices  for  reading  resistances  directly.  Ohmmeters  are  of  two 
general  types:  those  which  read  a  resistance  by  the  deflection  of 
a  pointer  over  a  scale,  this  deflection  being  proportional  to  the 
resistance,  but  independent,  within  wide  limits  of  the  value  of  the 
testing  current  employed;  and  those  which  operate  on  the  prin- 
ciple of  a  slide-wire  bridge,  which  is  balanced,  the  balanced  con- 
dition being  indicated  by  a  galvanometer  or  a  telephone.  The 
scale  under  the  slide  wire  is  laid  off  in  ohms  so  that  the  resistances 
are  read  directly  in  ohms. 

The  first  type  is  constructed  by  the  maker  so  as  to  operate 
directly  by  simply  connecting  the  resistance  to  be  read  to  two  bind- 
ing posts  and  reading  the  deflection  of  a  pointer.  A  description 
of  them  involves  describing  not  methods  of  measurement,  but  in- 
struments, and  they  will  not  be  further  considered  here. 

The  second  type  should  be  included  under  a  description  of 
"  balance "  methods  of  measuring  resistances. 

" Megger"  is  a  trade  name  given  to  a  type  of  deflection  ohm- 
meter,  which  is  very  fully  described  in  trade  publications.  It  is 
claimed  that  they  give  all  results  which  can  be  obtained  with  volt- 
meters in  the  measurement  of  resistances  and  with  greater  facility 
and  precision.  This  claim  is  probably  justified.  If  such  an  in- 
strument is  available  it  should  be  used  in  preference  to  a  voltmeter 
for  many  kinds  of  resistance  measurements. 

The  " Megger"  was  designed  and  patented  by  Mr.  Sidney 
Evershed  of  London,  for  the  rapid  determination  of  high  resist- 
ances. Following  is  a  very  brief  description  of  this  instrument 
and  some  of  the  claims  made  for  it. 

It  is  a  direct-reading  ohmmeter  with  a  direct-current  hand- 
driven  dynamo  mounted  in  the  same  case.  The  instrument  is 
arranged  for  easy  portability  and  rough  usage.  The  scale  is 
graduated  in  ohms  so  that  no  calculations  are  required.  The 
hand-driven  dynamo  delivers  current  at  100,  250,  or  1000  volts 
(according  to  the  range  and  sensibility  of  the  particular  type  used) 
and  this  makes  the  instrument  independent  of  outside  current 
supply.  The  instrument  is  similar  in  principle  to  a  differential 
galvanometer  so  devised  that  moderate  change  of  dynamo  voltage 
does  not  vary  the  scale  reading.  The  entire  instrument  weighs 
about  20  Ibs.  It  is  claimed  to  have  a  range  from  0.01  to  2000 
megohms.  Two  scales  of  high-range  meggers  are  shown  full  size 


ART.  2111        VOLTMETER  AND  AMMETER  METHODS 


39 


in  Fig.  211.  From  an  inspection  of  these  scales  one  may  estimate 
the  possible  precision  attainable  on  the  assumption  that  the  instru- 
ment itself  is  accurately  calibrated. 


FIG.  211. 


The  instrument  is  sold  in  America  by  James  G.  Biddle,  Phila- 
delphia, Pa.,  and  is  fully  described  in  his  catalogue  No.  740. 
This  being  an  instrument  rather  than  a  method  of  measurement, 
the  author  must  refer  the  reader  to  the  above-mentioned  pamphlet 
for  a  more  detailed  description. 


CHAPTER  III. 

NULL  METHODS.     RESISTANCE  MEASURED  BY 
DIFFERENTIAL   INSTRUMENTS. 

300.  Remarks  on  Null  Methods.  —  When  some  quantity  of 
known  value  is  varied  in  a  known  way  until  it  is  made  equal  to, 
or  to  some  known  multiple  of,  a  quantity  being  measured,  and 
when  the  attainment  of  this  condition  is  indicated  by  some  de- 
tector then  showing  no  deflection,  the  measurement  is  said  to  be 
made  by  a  "null"  or  "balance"  method.  The  most  familiar 
example  of  the  above  is  weighing  with  a  chemical  balance.  Here 
a  mass  is  the  unknown  quantity.  Known  masses  are  placed  in 
the  pan  until  the  balance,  which  is  here  the  detecting  instrument, 
shows  no  deflection.  Other  examples  are:  The  measurement  of 
resistance  with  a  Wheatstone  bridge,  where  known  resistances  are 
varied  until  they  are  made  a  known  multiple  of  an  unknown 
resistance.  The  zero  deflection  of  a  galvanometer  indicates  in 
this  case  when  the  known  resistances  have  been  perfectly  adjusted; 
the  measurement  of  potential  differences  with  a  potentiometer, 
where  a  known  potential  difference  is  varied  in  a  known  manner 
until  it  equals  an  unknown  potential  difference  under  measure- 
ment. The  zero  deflection  of  a  galvanometer,  the  detecting  in- 
strument, shows  in  this  case  also  when  the  equality  is  attained. 

Null  or  balance  methods  of  measurement  are  generally  more 
sensitive  and  accurate  than  deflection  methods:  1st,  because  it 
is  unnecessary,  as  in  deflection  methods,  to  know  the  constant  of 
the  detector;  2d,  because  it  is  unnecessary  to  know  the  law  of  the 
deflection  of  an  indicating  instrument,  or  to  rely  upon  the  accuracy 
or  the  constancy  of  the  calibration  of  a  scale;  3d,  because  quanti- 
ties of  the  same  kind  are  simply  matched  against,  or  compared 
with,  one  another;  4th,  because  it  is  easier  to  detect  a  small 
departure  from  the  zero  of  an  indicating  instrument  than  it  is  to 
read  the  exact  extent  of  a  deflection. 

For  reasons  of  the  above  character,  weighing,  bridge  methods 
of  measuring  resistance,  and  potentiometer  methods  of  measuring 
potential  differences  are  the  most  accurate  used. 

40 


ART.  301]  NULL  METHODS  41 

Balance  or  null  methods  of  measurement  have,  however,  the 
disadvantages  of  requiring  more  apparatus,  of  consuming  more 
time,  and  of  requiring  more  manual  manipulation  than  do  deflec- 
tion methods.  If  the  quantity  under  measurement  is  subject  to 
fluctuations,  which  it  is  desirable  to  note,  then  strictly  null  methods 
are  less  suitable  than  deflection  methods.  In  case  the  quantity 
being  measured  is  fluctuating,  it  is  generally  impossible  to  perform 
the  manual  manipulations  with  sufficient  rapidity  to  maintain  the 
balanced  condition  and,  even  if  this  can  be  done,  to  record  the 
settings. 

There  are  methods,  which  we  shall  discuss  later,  which  measure 
the  quantity  by  approximately  matching  it  with  a  known  quantity, 
deflections  being  used  to  determine  the  small  differences.  These 
balance-deflection  methods  are  of  great  value  in  industrial  elec- 
trical measurements,  as  they  combine  precision  and  speed. 

The  strictly  null  methods  are  chiefly  valuable  in  connection 
with  standardization  measurements.  Those  which  are  useful  in 
the  measurement  of  medium  resistances  will  receive  full  attention. 
The  methods  to  be  discussed  now  are  those  which  make  use  of  a 
differentially  wound  galvanometer  and  those  which  employ  some 
form  of  the  network  of  resistances  generally  known  as  the  Wheat- 
stone  bridge. 

The  differential  methods  have  certain  advantages  in  some  im- 
portant commercial  applications  and  therefore  deserve  a  careful 
consideration. 

301.  Properties  of  Differential  Circuits.  —  Almost  any  instru- 
ment which  deflects  with  the  passage  of  an  electric  current  thru 
it  may  be  differentially  wound;  namely,  so  wound  with  a  double 
winding  that  a  current  thru  one  winding  is  exactly  neutralized,  in 
its  action  to  produce  a  deflection,  by  an  equal  current  thru  the 
other  winding.  A  differential  winding  may  be  applied  to  milli- 
ammeters,  voltmeters,  galvanometers  of  the  moving-magnet  or 
of  the  moving-coil  type,  and  to  telephones.  When  applied  to 
direct-current  instruments  the  circuits  have  certain  interesting 
and  useful  properties,  to  a  consideration  of  which  we  now  pro- 
ceed. The  properties  to  be  discussed  are  quite  independent  of 
the  type  of  the  differential  instrument.  This  may  be  a  moving- 
magnet  or  a  moving-coil  galvanometer,  but  for  definiteness  let 
us  fasten  our  attention  upon  a  differentially  wound  D' Arson  val 
galvanometer. 


42 


MEASURING  ELECTRICAL  RESISTANCE         [ART.  301 


One  assumption  must  always  be  made,  and  a  second  assumption 
generally.  The  first  is  that  equal  currents,  but  opposite  in  their 
electromagnetic  action,  in  the  two  windings  shall  be  without  influ- 
ence upon  the  movable  system.  The  second  is  that  the  resistances 
between  binding  posts  of  the  two  windings  shall  be  the  same.  A 
third  assumption  is  sometimes  required  for  certain  classes  of 
refined  work.  It  is,  that  the  resistances  oT  the  two  windings 
shall  remain  constant  under  temperature  changes  in  the  room 
and  with  varying  quantities  of  current  in  the  windings.  This 
last  assumption  can  only  be  met,  as  a  rule,  with  difficulty  or  at 
the  sacrifice  of  sensibility,  as  when  coils  are  wound  with  wire  of 
high  resistance  to  secure  a  negligible  temperature  coefficient. 


II 


FIG.  301a. 


To  test  if  the  instrument  is  truly  differential  join  the  two  wind- 
ings in  series  as  shown  at  I,  Fig.  301a.  Then,  if  the  number  of 
turns  in  the  windings  gi  and  g%  are  equal  and  wound  in  opposite 
directions,  the  same  current  traversing  these  windings  should 
produce  no  turning  moment  on  the  system.  It  is  found  in  prac- 
tice, however,  that  howsoever  carefully  the  coil  is  wound  to  be 
made  differential,  some  tendency  will  remain  for  the  system  to 
turn.  Suppose  the  winding  gz  slightly  predominates.  If  a  very 
high  resistance  S  is  shunted  around  this  winding,  it  may  be 
adjusted  until  the  system  is  entirely  without  tendency  to  rotate, 
whatever  value  the  current  7  has;  provided  this  current  is  not 
so  large  as  to  greatly  heat  the  windings. 


ART.  301] 


NULL   METHODS 


43 


Next  join  the  windings  in  parallel,  as  in  II,  Fig.  301a,  and  vary 
a  small  resistance  n  in  series  with  gi,  or  a  resistance  r2  in  series 
with  £/2,  until  ii  =  i2,  when  the  system  will  again  give  no  deflection. 
With  these  two  adjustments  it  is  possible  to  make  the  system  differ- 
ential and  to  give  the  two  windings  exactly  the  same  resistance  — 
namely,  the  total  resistance  from  post  1  to  post  2  will  equal  the 
total  resistance  from  post  3  to  post  4.  The  above  adjustments 
should  be  made  by  the  instrument  maker  and  are  explained  here 
only  to  show  their  possibility. 

If  the  windings  g\  and  g2  are  of  copper  and  differ  very  much  in 
resistance,  then  changes  in  temperature  will  destroy  the  relation 
between  the  resistance  of  #2  and  the  shunt  S  and  the  adjust- 
ment to  differentially  will  not  be  permanent.  With  proper 
original  construction,  however,  such  a  defect  should  not  exist.  . 

It  will  now  be  assumed  in  what  follows  that  the  instrument  is 
truly  differential  and  that  the  two  windings  from  binding  post  to 
binding  post  have  the  same  resistance.  It  cannot  be  assumed 
that  the  resistances  of  the  windings  will  remain  constant  when 
these  are  of  copper,  but  if  they  change  with  temperature  they  will 
change  alike;  because  the  two  windings  are  made  by  winding  a 
double  wire  and  are  therefore 
intimately  associated,  and  hence 
always  at  the  same  temperature. 

We  proceed,  first,  to  prove 
an  interesting  and  useful  prop- 
erty of  the  circuits  of  a  differen- 
tial galvanometer  when  these 
are  made  up  in  the  manner  in- 
dicated in  Fig.  301b.* 

Let  1,  2,  and  3,  4  be  the 
binding  posts  of  the  differential 
instrument  (made  strictly  dif- 
ferential by  means  of  adjust- 
ments as  described  above). 


•Is 

A/WW- 


0 
FIG.  301b. 


Let  g  and  g  be  its  two  windings  so  disposed  that  equal  currents 
i  and  i  flowing  in  the  direction  indicated  by  arrows,  produce  no 
deflection. 

Let  fk  be  a  uniform  slide-wire  resistance  along  which  a  sliding 


*  Described  by  author  in  the  Trans,  of  the  Electrochemical  Society,  May, 
909,  Vol.  XV,  p.  340. 


44  MEASURING  ELECTRICAL  RESISTANCE         [ART.  301 

contact  t  may  be  moved,  and  let  this  slide  wire  be  shunted  by  a 
resistance  S. 

Let  R  be  any  fixed  resistance  from  0  to  /  and  X  any  resistance 
from  0  to  A;  which  may  be  varied. 

Now,  it  is  evident  that  if  R  and  X  are  nearly  equal  there  will  be 
some  position  for  t  on  the  slide  wire  fk  such  that  the  currents  i 
and  i  thru  the  two  windings  will  be  equal  and  the  system  give  no 
deflection. 

If  I  is  the  length  of  fk  (and  is  proportional  to  its  resistance) ,  we 
wish  to  determine  how  the  distance  c  of  t  from  /  varies  when 
X  varies.  If  c  varies  uniformly  with  variations  in  X  we  have  a 
means  of  measuring  small  changes  in  X  with  high  precision. 

The  signification  of  all  the  symbols  used  will  be  easily  under- 
stood by  a  reference  to  Fig.  301b,  without  further  explanation. 

By  Kirchoff's  laws, 

i  —  ic  —  ia  =  0  are  the  currents  which  enter  and  leave  /,       (1) 

i  +  ia  —  id  =  0  are  the  currents  which  enter  and  leave  k,      (2) 

iX  +  idd  =  E  is  the  E.M.F.  in  circuit  OktQ,  (3) 

iR  +  icc  =  E  is  the  E.M.F.  in  circuit  OftO,  (4) 

i»S  -f  idd  —  icc  =  0  is  the  E.M.F.  in  circuit  fSkf.  (5) 

Subtracting  (4)  from  (3)  gives 

i  (X-R)+  idd  -  icc  =  0.  (6) 

From  (1)  ie  =  i  -  i8.  (7) 

From  (2)  id  =  i  +  ia.  (8) 

Putting  the  values  of  ic  and  id  in  (6)  gives 

i  (X  -  R)  +  di  +  dis  -  d  +  cia  =  0,  (9) 

or     is  (c  +  d)  +  i  (X  -R)  -  i  (c-d)=  0.  (10) 

Putting  the  values  of  ic  and  id  in  (5)  gives 
i8S  +  di  +  dis  —  ci  +  cia  =  0, 

.   i(c-d) 


Also  from  (10)  we  have 

.^tfr-cO-^-B).  (12) 


Equating  the  second  members  of  (11)  and  (12)  we  derive 
R(c  +  d  +  S)+S(c-d) 


ART.  302]  NULL  METHODS  45 

In  (13)  replace  d'by  its  value  I  —  c  and  we  obtain 

v_  2cS    ,  R(l  +  S)-lS  .    . 

~S  +  l^          S  +  l 
The  last  term  of  (14)  is  a  constant. 
Call  this  K  and  we  have,  finally, 

K.  (15) 


Differentiating  (15),  we  find 
dX=  2S 
8c  ~  S  +  l' 

OTdc  =  ^ldX.  (16) 

Eq.  (16)  shows  that  c  varies  proportionally  to  X  and  only 

S  +  l 
0  g    as  rapidly  as  X. 

^  o 

This  result  means  that  in  the  case  of  a  uniform  slide  resistance, 
the  slide  resistance  may  be  given  any  convenient  value  greater 
than  the  minimum  allowed.  It  may  then  be  shunted  to  give  the 
particular  resistance  range  desired  with  any  chosen  length  of 
slide  resistance. 

By  minimum  resistance  allowed,  we  are  to  understand  a  resist- 
ance which  is  large  enough  to  permit  a  balance  to  be  obtained  with 
the  sliding  contact  upon  the  slide  resistance  fk  when  X  changes 
from  its  least  to  its  greatest  value. 

For  example,  if  when  X  has  a  minimum  value  the  contact  t  is 
at  point  /  for  a  balance,  it  must  be  possible  to  obtain  a  balance 
with  the  contact  t  at,  or  to  the  left  of,  the  point  k  when  X  has  its 
greatest  value. 

Putting  Eq.  (16)  in  the  form 


we  see  that  when  S  is  infinity,  that  is,  when  the  shunt  is  absent, 
the  variations  in  c  are  one  half  as  great  as  the  variations  in  X. 

302.  Illustration  of  the  Practical  Advantages  of  Differential 
Circuits.  —  To  see  the  advantages  of  the  above  properties  of 
differential  circuits  let  the  diagram,  Fig.  301b,  be  reconstructed 
as  given  in  Fig.  302. 


46 


MEASURING  ELECTRICAL  RESISTANCE         [ART.  302 


In  Fig.  302,  1,  2,  3  are  three  binding  posts  of  the  differential 
apparatus  and  4,  5  are  the  terminals  of  a  resistance  X  which  is 
supposed  to  be  placed  at  a  distance,  which  may  be  very  great, 
from  the  differential  instrument.  If  the  resistance  of  the  lead  wire 
A  is  made  equal  to  the  resistance  of  the  lead  wire  B,  Eq.  (16)  (§  301) 
still  holds.  For,  giving  A  a  resistance  and  B  an  equal  resistance 
is  not  different  from  increasing  by  an  equal  amount  the  resistance 
of  each  of  the  windings  g  and  g,  and  as  the  resistance  of  these 
windings  does  not  appear  in  either  Eq.  (15)  or  Eq.  (16)  (§  301)  the 


J 

o 
0 
o 

k 

ir 

3         jl 
A                         4          X 

FIG.  302. 

equal  resistances  A  and  B  do  not  enter  these  equations.  In  other 
words  the  resistance  X,  and  changes  in  the  resistance  X,  may  be 
determined  by  knowing  the  constant  resistances  R,  I,  S  and  the 
resistance  c  which  is  varied  for  obtaining  a  balance;  the  resistances 
g  and  g}  A,  B,  and  C  not  appearing. 

The  chief  industrial  application  of  the  above  principles  is  in 
the  measurement  of  temperature  with  electrical-resistance  ther- 
mometers which  may  be  placed  at  a  distance  from  the  reading 
instrument.  The  resistances  of  the  lead  wires  —  which  only  need 
to  be  three  in  number — can  vary  (provided  the  A  and  B  leads  are 
equal  in  resistance)  without  affecting  the  accuracy  with  which  the 
resistance  of  the  thermometer  is  determined,  and  hence  the  tem- 
perature which  is  a  function  of  its  resistance. 

Again,  the  method  may  be  employed  for  the  determination  of 
any  unknown  resistance  X.  If  the  contact  p  is  set  at  the  middle 
of  slide  resistance  fk  and  the  resistance  R  is  varied  until  a  balance 


ART.  302]  NULL   METHODS  47 

is  obtained,  we  have,  without  regard  to  the  resistance  of  the  lead 
wires,  X  =  R. 

If  R  is  made  equal  to  X  at  a  particular  temperature,  then 
small  variations  in  X,  due  to  temperature  changes,  can  be  accu- 
rately determined  by  moving  p  over  the  slide  resistance  fk  till  the 
galvanometer  is  balanced.  The  variations  in  X  are  then  given 

by  the  relation 

2  SI 

sx  =  s+ls-  .    ..     (« 

If  fk  is  a  slide  wire  of  uniform  resistance,  the  variations  in  c 
can  be  read  directly  on  a  millimeter  scale  and  thus  the  curve  giving 
the  relation  between  the  resistance  and  the  temperature  of  a  wire 
can  be  very  accurately  determined.  As  S  can  be  given  any  value, 
let  it  equal  infinity,  then  dX  =  2  be.  This  shows  that  when  the 
method  is  used  for  determining  temperature  coefficients,  the  highest 
value  of  c  (which  cannot  exceed  that  of  I)  will  not  be  greater  than 
one  half  the  total  change  which  takes  place  in  X  with  the  greatest 
variation  in  temperature  employed.  For  example,  suppose  it  is 
required  to  determine  the  temperature  coefficient  of  a  copper  coil 
which  has  a  resistance  of  10  ohms  at  0°  C.  in  the  range  0°  to  100°  C. 
At  100°  C  the  resistance  of  the  copper  coil  would  be  about  14.2  ohms. 
Then  its  increase  in  resistance  is  4.2  ohms  and  the  resistance  of  the 
slide  wire  fk  could  not  be  less  than  2.1  ohms.  The  resistance  R 
would  be  so  chosen  that  the  galvanometer  would  be  balanced 
when  X  was  at  0°  C.  and  p  is  set  at  /,  or  when  c  =  0.  Let  it  next 
be  required  to  determine  the  temperature  coefficient  of  a  1-ohm 
coil  in  the  same  range.  The  increase  in  resistance  of  this  coil  in 
the  100°  C.  range  would  only  be  0.42  ohm  and  the  distance  to 
move  on  the  slide  wire  would  only  correspond  to  0.21  ohm  or  0.1 
of  its  length.  If,  however,  the  slide  wire  is  shunted  with  a  resist- 
ance of  a  proper  value  (and  R  is  always  so  adjusted  that  with  the 
1-ohm  coil  at  0°  C.  we  have  c  =  0  f or  a  balance),  we  can  again  have 
c  =  I  when  the  1-ohm  coil  is  at  100°  C.,  and  thus  determine  the 
temperature  coefficient  of  1  ohm  with  the  same  percentage  pre- 
cision as  10  ohms. 

To  find  the  proper  value  to  give  S  we  assume  an  approximate 
increase  in  the  resistance  of  the  1-ohm  coil  when  the  temperature 
increases  100°  C.  We  then  solve  Eq.  (1)  for  S  and  find 


48 


MEASURING  ELECTRICAL  RESISTANCE         [ART.  303 


Since  5X  is  to  be  0.42  ohm,  and  I  has  previously  been  made  2.1 
ohms,  and  8c  is  to  be  practically  the  whole  length  of  the  slide 
wire,  or  2.1  ohms,  we  derive 


It  should  be  noted  in  the  above  methods  that  the  only  movable 
contact  is  in  the  battery  circuit,  and  hence  variations  in  its  resist- 
ance in  no  wise  affect  the  readings. 

Analysis  shows  that,  for  obtaining  greatest  sensibility,  each  wind- 
ing of  a  differential  instrument  should  have  a  resistance  which  is 
approximately  equal  to  the  resistance  external  to  that  winding. 
Or,  approximately,  we  should  have 

V      I  /'OX 

9  =  x  +  2 '  (3) 

303.  Differential  Galvanometer  Used  with  Shunts.  —  This 
method  is  perhaps  better  applied  with  a  differential  galvanometer 


FIG.  303. 

of  the  moving-magnet  type.  Some  of  these  galvanometers  are 
made  with  a  bell-shaped  magnet  hanging  in  a  hole  in  a  copper 
sphere  to  damp  the  oscillations  of  the  system.  The  current 
coils  are  two  in  number  and  can  be  individually  moved  along  a 
horizontal  bar  so  as  to  approach  or  recede  from  the  system. 
When  the  same  or  equal  currents  are  sent  in  opposite  directions 
thru  the  two  coils,  one  or  both  of  these  can  be  moved  into  a 
position  where  the  galvanometer  is  perfectly  differential.  If  this 
adjustment  is  made  and  it  be  assumed  that  the  resistance  of  its 
two  coils  are  alike  and  known,  and  if  they  can  be  relied  upon  to 
remain  constant,  we  can,  by  the  following  well-known  method, 
measure  any  resistance  X.  Let  the  circuits  be  arranged  as  in 
Fig.  303. 


ART.  303]  NULL   METHODS  49 

Here  g\  and  g%  are  the  two  windings  or  coils  supposed  to  be 
alike  in  resistance,  so  gi  =  g2  =  g,  the  resistance  of  each.  Each  of 
these  coils  is  shunted  with  resistances  Si  and  S2  respectively. 
R  is  a  fixed  resistance  and  X  a  resistance  to  be  measured. 

Denote  by  /  the  current  in  the  battery  circuit  and  let  i%  denote 
the  current  in  the  right-hand  coil  and  ii  the  current  in  the  left- 
hand  coil.  To  find  an  expression  for  the  value  of  X,  when  the 
galvanometer  is  balanced,  we  make  use  of  the  following  law  of  the 
division  of  currents: 

Let  a  circuit  divide  into  two  branches  of  resistances  x  and  y, 
and  let  the  resistance  x  carry  a  current  ii  and  the  resistance  ?/,  a 
current  i'2  and  let  ii  +  iz  =  I  be  the  total  current;  then 

77  '7* 

ii  =  —  7—  /    and    i'2  =  -      —  /; 

x+y  x+y 

that  is,  the  current  in  either  branch  is  equal  to  the  resistance  of 
the  other  branch  divided  by  the  total  resistance  around  the  circuit 
times  the  total  current. 

Applying  this  principle  to  the  circuits  shown  in  Fig.  303,  we 
have 

Y  [      s*9* 

j  _  &+02  & 


Y  _L  P  -4-         li      J_ 
•A-    i   ft  ~t~  TY     i      "  ~r 
oi-T0i 

and 


^2^2 
T~ 


•  Q  Q     _|_ 

01  +  91          &2  +  ^2 

If  the  galvanometer  is  so  adjusted  that  g\  =  gi  =  g,  and  we 
alter  Si,  S2,  and  R  until  there  is  no  deflection,  we  get  from  Eqs.  (1) 
and  (2) 


(3) 


^*  £t 

The  method  expressed  by  equation  (3)  can  only  give  accurate 
results  on  the  assumption  that  the  resistance  g  is  either  the  same 
at  different  temperatures,  or  is  known  at  the  temperature  at  which 
the  measurement  is  made.  To  realize  the  first  assumption  the 


50  MEASURING  ELECTRICAL  RESISTANCE         [ART.  304 

windings  must  be  made  of  some  low-temperature  coefficient 
wire,  as  manganin,  which  has  a  high  resistance,  and  to  realize 
the  second  assumption  requires  the  taking  of  accurate  temperature 
readings  of  the  coils.  Moreover,  it  would  in  this  case  be  impossible 
to  make  the  shunts  Si  and  S2  bear  any  fixed  relation  to-  the  re- 
sistance of  the  windings.  From  these  considerations  the  method 
is  seen  to  be  inferior  to  the  Wheatstone-bridge  methods  of  measur- 
ing resistance  which  are  in  vogue.  But,  with  galvanometer 
windings  of  manganin  and  shunts  of  suitable  values,  the  precision 
and  the  range  of  resistance-measurement  possible  can  be  made 
equal  to  that  of  the  ordinary  post-office  type  of  Wheatstone 
bridge.  For  example,  suppose 

Si  =  jy    and  S2  =  infinity, 

then,  by  Eq.  (3),  X  =  100  R,  and  if  R  can  be  varied  in  steps  of 
1  ohm  from  0  to  10,000  ohms  we  can  measure  a  resistance  X  = 
1,000,000  ohms.  Or,  suppose 

^2  =  QQ  anc*  $1  =  infinity? 

then,  by  Eq.  (3),  X=  0.01  R,  and  if  R  =  1  ohm,  X  =  0.01  ohm. 

The  above  method  is  thus  seen  to  require,  for  precision  and 
range,  a  specially  constructed  galvanometer  and  shunts  which 
have  values  of  at  least  ^  and  gV  of  the  resistance  of  a  coil  winding 
and  a  10,000-ohm  rheostat  variable  in  steps  of  1  ohm.  It  has, 
therefore,  little  to  recommend  it  in  competition  with  the  bridge 
methods  to  be  described,  and  is  given  here  chiefly  to  complete 
the  treatment  of  the  differentially  wound  instrument  as  used  for 
resistance-measurements . 

304.  The  Differential  Telephone.  —  The  magnet  of  a  tele- 
phone may  be  wound  with  a  differential  winding.  This  instru- 
ment can  then  be  used  to  indicate  when  there  is  an  equality 
between  two  currents  which  are  alternating  or  unsteady.  The 
two  wires  of  the  differential  winding  should  be  twisted  together 
and  wound  bifilar  upon  the  magnet.  The  theory  of  this  instru- 
ment is  complicated,  and  the  circumstances  under  which  it  can 
be  used  to  advantage  are  limited,  therefore  we  shall  not  give 
further  discussion  to  it  here. 


CHAPTER  IV. 


THE  WHEATSTONE-BRIDGE  NETWORK. 
BRIDGE  METHODS. 


SLIDE-WIRE 


400.  Network  of  the  Wheatstone  Bridge.  —  The  Wheatstone 
bridge  is  a  network  of  six  conductors  and  should  be  distinguished 
from  the  Kelvin  double  bridge  (to  be  discussed  later)  which  is  a 
network  of  nine  conductors.  While  the  Wheatstone  net  or  bridge 
may  assume  many  forms,  the  essential  electrical  properties  are 
the  same  in  all. 

This  network  of  six  conductors  may  be  represented  in  three 
ways  which  are  equivalent.  They  are  presented  in  I,  II,  and  III, 
Fig.  400a. 


In  the  three  representations  like  letters  designate  the  same 
conductors  and  the  same  points  of  junction.  Following  the 
lettering  of  the  diagrams,  it  will  be  seen,  from  a  well-known  prop- 
erty of  the  bridge,  that,  if  the  resistances  6,  c,  x,  y  have  the 
relation 

yb  =  ex  (1) 

there  will  be  no  current  in  a  if  there  is  an  E.M.F,  in  z,  and,  con- 
versely, there  will  be  no  current  in  z  if  there  is  an  E.M.F.  in  a. 
Under  these  circumstances  the  two  conductors  a  and  z  are  said 
to  be  conjugate  conductors.  Other  pairs  of  conductors  can  become 

51 


52 


MEASURING  ELECTRICAL  RESISTANCE         [ART.  400 


conjugates.     By  referring  to  diagram  I  we  can  write,  from  the 
symmetry  of  the  diagram,  the  following  relations: 

If  yb  =  ex,  then  z  and  a  are  conjugates. 

If  za  =  by,  then  x  and  c  are  conjugates. 

If  xc  =  az,  then  y  and  6  are  conjugates. 

The  above  relations  belong  to  what  we  shall  term  the  "  first  prop- 
erty "  of  the  Wheatstone  bridge. 

We  shall  presently  show  that  if  any  two  conductors  are  con- 
jugates, in  whatever  part  of  the  network  an  E.M.F.  be  placed,  the 
current  which  flows  through  one  of  the  conjugates  is  independent 
of  the  resistance  of  the  other  conjugate.  For  example,  if  in 
diagram  III  a  and  z  are  conjugates  and  an  E.M.F.  is  placed  in 
branch  c,  the  current  which  will  flow  in  z  will  be  the  same  what- 
ever may  be  the  resistance  of  its  conjugate  a.  This  principle 
is  made  use  of  later  in  determining  the  internal  resistance  of  a 
battery  and  the  resistance  of  a  galvanometer.  We  shall  term  this 
the  "  second  property  "  of  the  Wheatstone  bridge.  The  above- 
mentioned  "  first  "  and  "  second  "  properties  of  the  Wheatstone 
bridge  may  be  deduced  as  follows: 

A 


FIQ.  400b. 

If,  in  Fig.  400b,  we  place  an  E.M.F.  in  the  branch  BC  we  shall 
have  the  distribution  of  currents  shown  in  the  diagram,  where 

ie  is  the  current  in  c, 

ia  is  the  current  in  a, 

ix  is  the  current  in  x, 

ie  —  iy  is  the  current  in  y, 

ia  —  ix  is  the  current  in  6, 

and  ic  —  iy  +  ix  is  the  current  in  z. 

As  before  (Fig.  400a),  x,  y,  z,  a,  b,  c  designate  ohmic  resistances. 
If  it  is  true  that  x  and  c  are  conjugates,  then  the  E.M.F.  in  c  will 


ART.  400]         THE  WHEATSTONE-BRIDGE   NETWORK  53 

produce  no  current  in  x,  and  hence  we  shall  have  ix  =  0.  Assume, 
then,  the  current  ix  is  zero  and  determine  the  relation  of  the  re- 
sistances necessary  to  produce  this  result.  With  ix  =  0,  the  points 
A  and  0  will  be  at  the  same  potential,  and  we  shall  have  by  Ohm's 
law  the  fall  in  potential  from  B  to  A  =  aia  and  the  fall  in  potential 
from  B  to  0  =  y  (ic  —  iv)  and  these  will  equal  each  other,  or 

aia  =  y  (ic  ~  iv).  (2) 

Likewise  (remembering  that  we  have  assumed  ix  =  0)  we  shall 
have 

bia  =  z  (ic  -  iy).  (3) 

Hence,  taking  the  ratio  of  Eq.  (2)  to  Eq.  (3),  we  obtain 

a     y 

1  =  -,   °r    za  =  by, 

which  is  the  relation  necessary  to  make  x  and  c  conjugates.  The 
above  proves  the  "  first  property  "  of  the  bridge. 

To  prove  the  "  second  property  "  assume  that  an  E.M.F.  exists 
in  some  branch  of  the  bridge,  as  AC  (Fig.  400b).  Then  a  cur- 
rent will  flow  in  the  branch  BC  as  well  as  in  the  branch  AO. 
Assume  now  a  counter  E.M.F.  to  be  introduced  into  the  branch 
BC.  This  E.M.F.  cannot  produce  any  current  in  the  conjugate 
AO.  In  particular  choose  this  counter  E.M.F.  such  as  to  just 
reduce  the  current  in  BC  to  zero.  But  when  an  E.M.F.  is  intro- 
duced into  BC,  which  reduces  the  current  in  this  branch  to  zero, 
we  have  done  the  equivalent  of  opening  the  circuit  BC.  Hence 
it  follows  that  the  current  in  AO  resulting  from  an  E.M.F.  in 
AC  is  unaffected  by  any  change  in  the  resistance  of  BC. 

The  mathematical  relations  which,  hold  for  the  Wheatstone 
bridge  when  this  is  unbalanced,  namely,  when  no  two  conductors 
are  conjugates,  are  complicated  and  require  lengthy  calculations 
to  derive.  These  relations  are  given  in  such  standard  works  as 
Kempe's  "  Handbook  of  Electrical  Testing."  They  are  rarely 
made  use  of  in  the  practical  employment  of  the  Wheatstone  bridge, 
and  we  do  not  consider  that  it  would  be  advisable  to  give  a  general 
treatment  of  these  relations  here.  We  shall,  therefore,  proceed 
at  once  to. a  consideration  of  those  features  which  are  useful  to 
know  and  understand  in  applying  the  Wheatstone  bridge  to  such 
electrical  measurements  as  arise  in  practice. 

Represent  the  bridge  as  in  Fig.  400c.  Here  KI  and  K2  are  the 
two  conductors  which  are  to  become  conjugates  when  the  bridge 


54 


MEASURING  ELECTRICAL  RESISTANCE         [ART.  401 


is   balanced.     If  a,  6,  c,  d  represent  the  resistances  of  their  re- 

a      c 
spective  arms,  this  condition  is  filled  when  T  =  j  •     If  any  three  of 

these  four  resistances  are  known  the  other  is  given  by  this  relation. 

In  conformity  with  custom  we  shall 
call  the  two  adjacent  arms  which 
do  not  contain  the  unknown  resist- 
ance the  "  ratio  arms "  and  the 
arm  adjacent  to  the  unknown  re- 
sistance the  "  rheostat "  of  the 
bridge.  It  is  evident  that  the  bridge 
may  be  balanced  either  by  varying 
the  ratio  arms  or  by  maintaining 
the  latter  fixed  and  varying  the 
rheostat.  When  the  former  plan  is 
followed,  the  bridge  usually  takes 

the  form  of  a  so-called  slide-wire  or  slide-resistance  bridge  and 
when  the  latter  plan  is  followed  the  bridge  is  usually  a  so-called 
plug-type  or  dial-type  bridge. 

401.  Uses  of  the  Slide-wire  Bridge.  —  The  slide-wire  bridge 
(often  referred  to  as  the  "  meter  bridge  "  because  a  slide  wire  a 
meter  long  stretched  over  a  meter  scale  is  used)  was  one  of  the 
earliest  if  not  the  earliest  form  of  a  Wheatstone  bridge. 


FIG.  400c. 


7H 

n 

R 

n 

Ga          rvwv-j 

r 

i 

h 

c 

1  —  n_        ^ 

1        l- 

2 

C 

A 

7              P 

J 

7>  , 

fci 

t     ~ 

FIG.  401a. 

»j 

This  form  of  the  Wheatstone  bridge  is  diagrammatically  shown 
in  Fig.  401a.  Here  R  is  a  known  resistance,  which  may  be  given 
various  convenient  values,  x  is  the  resistance  to  be  measured  and  n\ 
and  ri2  are  two  resistances  which  may  be  inserted  at  the  ends  of  the 
slide  wire. 

To  use  this  bridge,  a  resistance  R  is  chosen  which,  preferably, 
is  as  nearly  as  possible  equal  to  the  resistance  x  to  be  measured. 


ART.  401]        THE  WHEATSTONE-BRIDGE  NETWORK  55 

The  resistances  n\  and  n%  should  be  chosen  low  enough  so  that  the 
bridge  may  be  balanced  by  sliding  the  sliding  contact  p  to  some 
point  on  the  bridge  wire.  The  battery  and  galvanometer  may 
be  located  as  shown  in  the  diagram  but  the  position  of  these  may 
be  reversed.  If  I  is  the  length  of  the  bridge  wire  and  c  the  distance 
from  A  to  p  in  millimeters,  and  if  n\  and  nz  are  resistances  deter- 
mined in  equivalent  millimeters  of  length  of  bridge  wire  we  have  for 
a  balance 


x.  ".  n2+  /  —  c' 

*  =  ??t±l=-Cfl.  '(1) 

ni  +  c 

In  practice  the  two  resistances  n\  and  n2  would  be  made  equal 

and  may  then  be  called  rc,  and  I  would  be  1000  mm. 

Then 

g  =  n  +  1000-e 

n  -f  c 

The  value  which  may  be  given  to  the  resistances  n  and  n  will 
depend  upon  how  nearly  it  is  possible  to  choose  the  resistance  R 
to  be  like  the  resistance  x.  Since  the  greatest  value  c  can  have 
is  1000,  and  a  balance  still  be  obtained  with  the  slider  upon  the 
wire,  we  assume  c  to  have  this  value  and  solve  Eq.  (2)  'for  n  and 
thus  find, 

1000  x  ,_. 

n  =  /T=V  <3> 

as  the  maximum  value  n  may  have.    Suppose  it  is  possible  to 
always  choose  R  so  that  the  value  of  the  resistance  x  which  we 

D 

have  to  measure,  will  never  be  less  than  ^  .     Then  we  could  make 


Namely,  n  could  equal  the  resistance  of  1000  mm  of  bridge  wire. 
If  the  resistance  R  is  obtained  from  a  small  plug  resistance  box, 
this  is  almost  always  practicable.  Using,  then,  this  value  of  n, 
Eq.  (2)  becomes 

_  2000  -  c 

'1000  +  c 


56  .     MEASURING  ELECTRICAL  RESISTANCE         [ART.  401 

The  effect  of  using  resistances  n  and  n  each  of  which  are  equal  in 
resistance  to  1000  mm  of  bridge  wire,  is,  virtually,  to  make  this 
wire  three  times  as  long.  Furthermore,  by  care  in  construction,  the 
resistances  n  and  n  may  be  made  to  include  the  unavoidable  resist- 
ances of  the  joints  and  copper  straps  between  the  points  e\  and 
0i,  and  e<i  and  g%  (Fig.  401a).  When,  as  in  the  more  ordinary 
use  of  the  slide-wire  bridge,  the  resistances  n  and  n  are  assumed 
equal  to  0,  and  the  formula 

(5) 


is  used,  errors  are  sure  to  result  from  a  neglect  of  the  small  and  un- 
known resistances  which  exist  between  the  galvanometer  termi- 
nals 0i,  02  and  the  ends  of  the  wire  e\,  e^  where  the  scale  begins  and 
ends.  The  use  of  the  resistances  n  and  n  limits  the  range  of  the 
bridge  (in  the  case  of  Eq.  (4),  from  x  =  2  R  to  x  =  0.5  R),  but  gives 
greater  precision  than  can  be  obtained  without  them.  When 
these  resistances  are  not  used  and  the  connecting  resistances  g\  to 
ei  and  02  to  62  are  kept  very  small  by  a  good  construction  of  the 
bridge,  the  range  of  measurement  is,  theoretically,  from  zero  to 
infinity,  tho  in  practice  the  precision  for  either  extreme  is  very 
low.  If  the  method  expressed  in  Eq.  (5)  is  used,  it  is  always 
desirable  to  choose  R  as  nearly  equal  to  x  as  possible  so  as  to  bring 
the  balance  point  near  to  the  center  of  the  wire.  It  should  be 
noted  that  Eq.  (5)  can  be  written, 

x  =  (reciprocal  of  scale  reading  X  1000  —  1)  R. 

Thus  by  using  a  table  of  reciprocals,  or  a  slide  rule,  the  values  of 
x,  when  a  number  of  measurements  are  to  be  made,  may  be  cal- 
culated simply  and  rapidly.  For  R  one  would  choose  values,  as 
1,  10,  100,  or  1000,  etc. 

In  the  above  methods  employment  is  made  of  a  slide  wire  of 
high  resistance  alloy  one  meter  in  length.  It  is  now  possible  to 
obtain  alloys  of  practically  zero  temperature  coefficient  and 
having  60  times  the  resistivity  of  copper.  If  the  slide  wire  is 
No.  24  B.  &  S.  gauge  its  resistance  per  meter  might  be  made 
about  5  ohms. 

It  is  very  desirable  to  be  able  to  produce  a  uniform  slide  resist- 
ance of  100  or  more  ohms.  This  is  accomplished  by  certain 
well-known  makers,  by  winding  on  a  long  mandril  of  about  3  mm 
diameter  an  insulated  resistance  wire  and  then  withdrawing  the 


ART.  401]        THE  WHEATSTONE-BRIDGE  NETWORK 


57 


mandril.     This  leaves  a  long  helix  of  insulated  wire  with  the  turns 
close  together.     This  helix  is  laid  in  a  groove  in  a  strip  of  hard 


M.F: 


Std.0ap 
or  f 
EetS 


XCap 
or 

>td. 


Phone 
or  f 


Spiral  of  Insulated  Wire 
Insulation  removed  where 
Contact  Bears 


FIG.  40 Ib. 

wood  or  hard  rubber,  or  in  a  groove  in  the  periphery  of  a  disk  of 
hard  rubber  and  cemented  in  place  with  thick  shellac.     When  the 


58  MEASURING  ELECTRICAL  RESISTANCE         [ART.  402 

shellac  is  dry  and  hard,  the  insulation  is  removed  along  one  side 
of  the  helix  parallel  with  its  axis.  A  sliding  contact  is  arranged 
to  slide  over  the  bared  portion  of  the  helix  so  as  to  make  contact 
at  any  point  along  its  length.  In  this  manner  a  substantial  slide 
resistance  may  be  made  which  can  be  given  a  resistance  of  as 
much  as  1000  ohms  to  the  meter.  By  rubbing  one  side  of  the  helix 
with  emery  cloth  such  a  slide  resistance  can  be  made  very  uni- 
form in  resistance  from  one  end  to  the  other. 

When  such  a  slide-wire  resistance  is  used,  the  bridge  wire  is 
usually  made  circular  in  form  and  the  contact  is  moved  around 
the  periphery  of  the  circular  disk  by  means  of  a  handle  at  its 
center.  The  disk  supporting  the  slide  wire  is  placed  underneath 
a  rubber  plate,  and  a  circular  scale  and  a  pointer  which  moves 
with  the  contact  are  placed  above  the  rubber  plate.  The  con- 
struction is  illustrated  in  Fig.  401b. 

If  the  scale  above  the  rubber  plate  is  properly  laid  off,  in  the 
manner  indicated  in  the  figure,  it  is  only  necessary  to  multiply 
the  scale  reading  by  the  value  of  the  fixed  resistance  R  which, 
being  chosen  1,  10,  100,  or  1000  ohms,  makes  the  instrument 
direct  reading.  Hence  it  becomes  a  slide-resistance  ohmmeter. 
When  mounted  with  a  small  portable  pointer  galvanometer  it 
becomes  a  portable  ohmmeter  and  is  a  convenient  laboratory  tool. 
The  instrument  as  regularly  made  is  capable  of  giving  measure- 
ments of  an  accuracy  of  0.25  of  1  per  cent  when  a  standard  is  chosen 
which  will  bring  the  reading  not  far  from  the  center  of  the  scale. 

In  the  classes  of  slide-wire  bridge  measurements  as  described 
above,  sufficient  sensibility  may  be  obtained  from  a  pointer  gal- 
vanometer of  from  30  to  100  ohms  resistance  of  the  types  made 
by  Paul,  of  London;  The  Leeds  and  Northrup  Company,  of 
Philadelphia,  Pa. ;  or  by  Edward  Weston. 

The  type  of  slide-wire  bridge  measurements,  described  in  con- 
nection with  Fig.  40 la,  with  a  portable  pointer  galvanometer  used 
as  a  detecting  instrument  is  very  suitable  for  instruction  in  the 
use  of  the  Wheatstone-bridge  principle  and  is  to  be  recommended 
for  college  laboratories. 

402.  Comparison  of  Resistances  by  Modified  Slide- wire 
Bridge.  —  This  method,  which  is  applicable  to  low-resistance 
conductors,  is  explained  as  follows  (Fig.  402) :  Let  1-2  be  a  linear 
conductor  of  uniform  resistance,  the  resistance  per  unit  length 
of  which  we  wish  to  determine,  and  let  3-4  be  a  linear  conductor 


ART.  402]         THE  WHEATSTONE-BRIDGE   NETWORK 


59 


of  uniform  resistance,  the  resistance  per  unit  length  of  which  is 
given.     Ri  and  R2  are  ratio  coils. 


-1  1  

J    a                              aL 

"^  U  '-'      ^                              b    l 
1  f 

|l4 
i 

Ba  K 

FIG.  402 


The  two  conductors  are  joined  together,  no  special  regard  being 
taken  to  make  this  a  low-resistance  contact.  The  battery  is 
joined  to  the  ends  1  and  4  with  a  key  K  in  circuit.  With  con- 
tact pi  set  at  a,  move  contact  p2  to  some  point  of  balance  b. 
Next  move  contact  pi  to  «i  which  is  some  accurately  measured 
distance  I  from  a.  Now  move  contact  p2  to  some  point  61  where 
a  balance  is  again  obtained  and  accurately  measure  or  read  off 
on  a  scale  beneath  the  conductor  the  distance  L  of  61  from  b. 
When  a  balance  is  first  obtained  the  relation  holds: 

R  i  _  resistance  a  to  0 

Rz      resistance  6  to  0* 
When  the  balance  is  next  obtained  the  relation  holds : 

Ri      resistance  a\  to  0 


Hence, 


atoO 
btoO 


resistance  61  to  0 

a  to  0 


bitoO' 
It  is  proved  in  algebra  that,  if 


or 


6  toO 


en  to  O      bi  to  0 


Now 

and 
hence, 


x      w       , ,          x  —  y      x 
-  —  — .     then  -  =  —  • 

y      z  w  —  z      w 

(a  to  0)  —  (ai  to  0)  =  (a  to  ai) 
(6toO)  -  (fcitoO)  =  (b  to  61); 


a  to 


a  to  0 


b  to  61  ~  6  toO 


60  MEASURING  ELECTRICAL  RESISTANCE         [ART.  402 

or,  *~rW-  (1) 

Itz 

Eq.  (1)  states  that  the  resistance  z  of  a  length  /  of  the  rod  1-2  is 

r> 

-^  times  the  resistance  r  of  a  length  L  of  the  rod  3-4.  If  this  last 
KZ 

is  known  the  other  is  given,  and  the  method  enables  one  to  compare 
in  a  very  simple  way  and  with  inexpensive  apparatus  the  resistance 
of  a  given  length  of  one  conductor,  as  a  rod  of  aluminum,  with  that 
of  another  conductor,  as  a  rod  of  copper. 

If  the  conductor  used  as  a  standard  has  the  same  temperature 
coefficient  as  the  conductor  which  is  being  compared  with  it,  no 
regard  has  to  be  taken  of  temperature  other  than  to  be  sure  that 
the  two  rods  are  at  the  same  temperature. 

With  a  D' Arson val  galvanometer  of  such  a  sensibility  that 
10~8  ampere  will  produce  a  deflection  of  one  scale-division  with 
the  scale  at  a  meter  distance,  as  the  detecting  instrument,  the  ratio 
coils  RI  and  Rz  may  assume  resistances  considerably  higher  than 
100  ohms,  so  that  the  contact  resistances  of  the  sliders  p\  and  p2 
can  be  entirely  neglected. 

This  method  gives  results  in  low-resistance  measurements 
similar  to  those  obtained  with  a  Kelvin  double  bridge  to  be  later 
described.  But  in  this  method  a  balance  must  be  obtained  twice, 
instead  of  only  once  as  with  the  Kelvin  double  bridge. 

The  method  is  very  conveniently  applied  when  the  conductor 
3-4  is  a  5-ohm  meter  bridge,  the  wire  of  the  meter  bridge  resting 
upon  a  meter  scale  marked  off  in  millimeters.  The  resistance  of 
this  wire  per  centimeter  of  length  must  be  accurately  known.  If 
the  conductor  1-2  is  of  low  resistance,  then  RI  would  be  made 
smaller  than  R2,  so  that  a  length  I  on  1-2  of,  say,  50  cms.  would 
correspond  in  resistance  with  a  length  of,  say,  75  cms.  on  the 
bridge  wire. 

This  method  may  likewise  be  applied  to  the  measurement  of 
low  resistances  between  potential  points.  In  this  case  pi  would 
first  be  set  on  one  potential  point,  and  a  balance  obtained  with 
pz  at  b,  then  on  the  other  potential  point  and  a  balance  be  again 

r> 

obtained  with  p2  at  61,  the  ratio  ^  being  so  chosen  that  L  would 

KZ 

be  a  considerable  proportion  of  the  length  of  the  bridge  wire. 
Then  if  x  is  the  resistance  sought  between  potential  points,  and 
if  p  is  the  resistance  of  the  bridge  wire  3-4  per  centimeter  of  length, 


ART.  403]         THE  WHEATSTONE-BRIDGE  NETWORK  61 

we  have  as  above, 

*  =  |LP,  (2) 

which  gives  x  in  ohms,  when  L  is  in  centimeters. 

403.  The  Carey-Foster  Method.  —  The  Carey-Foster  method 
of  using  a  slide-wire  bridge,  while  being  a  very  elegant  precision 
method,  is  not  as  much  employed  in  this  country  as  formerly. 
It,  nevertheless,  deserves  a  somewhat  extended  consideration. 
Tho  the  method  was  originally  devised  for  the  measurement  of 
very  low  resistances,  it  is  even  better  adapted  to  the  accurate 
comparison  of  medium  or  even  high  resistances.  The  success 
with  which  the  Carey-Foster  method  may  be  applied  to  precision 
work  in  comparing  resistances  depends  a  good  deal  upon  how  well 
the  apparatus  which  is  needed  for  carrying  out  the  method  is  de- 
signed and  made.  We  can  only  give  here  the  theory  of  the  method 
referring  the  reader  to  trade  publications  for  a  description  of  the 
mechanical  features  of  the  Carey-Foster  bridges  which  are  upon 
the  market. 

The  unique  feature  of  the  Carey-Foster  modification  of  the 
Wheatstone  slide-wire  bridge  consists  in  interchanging  the  stand- 
ard resistance,  which  is  in  one  arm  of  the  bridge,  and  the  resistance 
under  comparison  which  is  in  the  adjacent  arm.  A  reading  is 
taken  of  the  position  of  the  contact  on  the  slide  wire  necessary 
for  a  balance  before  and  after  the  resistances  are  interchanged. 
By  this  procedure  all  resistances  other  than  those  being  measured, 
as  well  as  all  constant  thermal  E.M.F.'s  in  the  bridge  circuits,  are 
eliminated. 

The  connections  for  the  Carey-Foster  bridge  method  are  shown 
in  Fig.  403a.  S  is  a  standard  resistance  coil  and  Si  is  a  coil  of 
approximately  the  same  value,  which  is  to  be  compared  with  S. 
Ri  and  R2  are  two  fixed  resistance  coils  of  nearly  equal  value. 
di  and  a2  are  the  number  of  units  of  length  of  the  portions  of  the 
bridge  wire  to  the  left  and  right  respectively  of  the  galvanometer 
contact  p.  If  p  is  the  resistance  of  one  unit  of  length  of  the 
bridge  wire,  then  pai  and  pa2  represent  the  resistance  of  the  bridge 
wire  to  the  left  and  to  the  right  of  the  galvanometer  contact. 
n  and  n\  may  be  taken  to  represent  all  resistances  of  unknown 
value  in  the  two  bridge  arms.  As  above  stated,  the  Carey-Foster 
method  provides  for  eliminating  these  unknown  resistances  by 
taking  two  readings  of  the  position  of  the  galvanometer  contact 


62 


MEASURING  ELECTRICAL  RESISTANCE         [ART.  403 


p  on  the  bridge  wire,  one  reading  when  the  coils  S  and  Si  have 
the  position  shown  in  Fig.  403a  and  one  when  these  coils  are  inter- 
changed. The  bridge  is  mechanically  so  constructed  that  this 
interchange  of  the  standard  coil  and  the  coil  under  comparison 
can  be  quickly  and  easily  made. 


The  dotted  line  a,  Fig.  403a,  represents  a  low-resistance  shunt 
to  the  bridge  wire  and  its  leads,  which  may  or  may  not  be  used. 
When  used  it  has  the  effect  of  magnifying  the  distances  that  the 
contact  p  must  be  moved  along  the  bridge  wire  to  obtain  a  balance 
in  the  two  positions  which  the  coils  S  and  Si  are  made  to  assume. 
The  general  theory  of  the  method  is  the  same  whether  this  shunt 
is  or  is  not  used. 

When  S  and  Si  have  the  positions  shown  in  Fig.  403a,  and  the 
contact  p  is  so  chosen  that  no  current  flows  in  the  galvanometer 
circuit, 

Ri=   S  +  n  +  pai  (} 

7~)       ~          o          [  j  ^  \      / 

where  p  is  the  resistance  per  unit  length  of  the  bridge  wire. 

S  and  Si  are  now  interchanged  and  a  balance  is  again  obtained 
by  sliding  p  to  a  new  position  on  the  bridge  wire.  Calling  the 
two  lengths  of  bridge  wire,  which  are  now  to  the  left  and  to  the 
right  of  the  slider  a/  and  a2', 

Ri  _  Si  +  n  -f  pai  /2\ 

7")  £f       |  j  /  \      / 

Changing  the  forms  of  Eqs.  (2)  and  (3)  by  adding  unity  to  each 
side,  we  obtain 

Ri  +  R2      S  +  Si  +  p  (di  +  a2)  +  n  +  ni 


pa2 


(3) 


ART.  403]         THE  WHEATSTONE-BRIDGE  NETWORK  63 

Ri  +  R2      S  +  Si  +  P  (ai  +  a*')  +  n  +  ni  , 

and  ""  ' 


Changing  the  position  of  the  galvanometer  contact  p  does  not 
alter  the  total  resistance  of  the  bridge  wire,  hence, 

P  (ai  +  a2)  =  P  (a/  +  a2')-  (5) 

The  numerators  of  the  second  members  of  Eqs.  (4)  and  (5)  are, 
therefore,  equal,  and,  as  the  second  member  of  each  of  the  two 
equations  is  equal  to  the  same  quantity,  the  denominators  of  the 
two  equations  are  equal. 
Thus,  we  have 

Si  +  paz  +  m  =  S  +  pat  +  ni, 
or 

-Si  =  S  -  P  (a,  -  a/),  (6) 

or,  as  paz  —  paz'  =  pai  —  pai  =  p  (a/—  ai), 

Si=>S-p  (a/  -  ai).  (7) 

Eqs.  (6)  and  (7)  show  that  the  difference  in  the  resistance  of  the 
standard  resistance  S  and  the  resistance  Si  under  comparison  is 
equal  to  the  resistance  of  a  certain  length  of  bridge  wire. 

If,  once  for  all,  the  resistance  p  per  unit  length  of  the  bridge 
wire  be  determined,  then  the  difference  in  the  resistance  of  any 
coil  under  comparison  and  a  standard  coil  is  given  in  ohms  with 
great  accuracy. 

There  are  several  methods  known  for  determining  the  value  of  p. 
A  simple  but  inferior  method  is  to  balance  two  standard  coils  the 
difference  of  whose  resistances  S  and  Si  is  accurately  known,  then 

S  —  Si  /0s 

P  =  —^  —  /•  (8) 

&2  —  #2 

On  account  of  differences  of  temperature  and  temperature 
changes  in  the  coils  S  and  Si  it  is  not  simple  to  accurately  deter- 
mine the  difference  in  their  resistances.  We  have,  therefore,  found 
the  following  method  of  determining  the  value  of  p,  by  means  of 
four  readings,  to  be  very  accurate  and  satisfactory  : 

It  is  necessary  for  the  determination  to  use  a  coil  Si  the  value 
of  which  does  not  need  to  be  known,  but  which  is  sufficiently 
near  the  resistance  of  coil  S  to  make  it  possible  to  obtain  a  balance 
with  p  (Fig.  403b)  not  far  from  the  center  of  the  bridge  wire.  An 
ordinary  resistance  spool  or  resistance  box  will  answer  very  well. 
The  resistance  may  be  varied  by  shunting  or  by  any  rheostat 


64 


MEASURING  ELECTRICAL  RESISTANCE         [ART.  403 


method  until  a  value  is  reached  by  trial  which  will  balance  the 
standard  S  with  p  near  the  center  of  the  bridge  wire. 

C  is  another  coil  or  box  of  resistance  coils  with  which  S  may  be 
shunted  by  closing  the  contact  K.  This  shunt  coil  should  have 
about  100  times  the  resistance  of  S  and  its  resistance  need,  not  be 

known  to  a  high  degree  of  accu- 
racy. 

The  four  readings  are  taken  as 
follows: 

Reading  1  is  taken  with  K 
open  and  S  and  Si  in  the  posi- 
tions shown  in  Fig.  403b.  Call 
this  reading  a\.  Reading  2  is 
taken  with  S,  together  with  coil 
C,  and  Si  interchanged  and  K 
open.  Call  this  reading  a2.  St 
together  with  coil  C,  and  Si  are 

now  interchanged  again  so  that  they  have  their  original  positions. 
K  is  now  closed,  shunting  S  with  (L     Call  the  resistance  of  S  when 
shunted  Si.      Reading  3  is  then  taken,  which  we  will  call  a/. 
Lastly  S  together  with  coil  C  is  interchanged  with  Si,  K  is  closed, 
and  reading  4  is  taken,  which  we  call  a* . 
The  value  of  p  is  then  deduced  as  follows: 
Before  S  was  shunted 


K 


FIG.  403b. 


After  shunting  S 


Si  =  S  —  p  (a2  —  ai). 

F 

Si  =  Sif  -p  (aj  -  a/) 


Eliminating  Si  from  Eqs.  (9)  and  (10),  we  derive 

Sif-S 


p  = 


0,1 


Since  Si'  = 


CS 


C  +  S 


,  the  value  of  p  may  also  be  expressed 


p  = 


(9) 
(10) 

(ID 
(12) 


When  low-resistance  coils  are  being  compared  the  distance  that 
the  galvanometer  contact  p,  Fig.  403a,  moves  over  the  bridge  wire 
in  obtaining  a  balance  with  S  and  Si  in  the  two  positions  is  often 
very  small. 


ART.  403]         THE  WHEATSTONE-BRIDGE  NETWORK  65 

In  order,  therefore,  that  the  same  slide  wire  may  serve  for  com- 
paring coils  of  both  high  and  low  resistance  it  has  been  found 
advantageous  to  be  able  to  shunt  the  bridge  wire  with  a  low-resist- 
ance shunt.  This  shunt  is  represented  in  dotted  line  in  Fig.  403a 
and  called  v. 

The  lower  the  resistance  of  this  shunt  the  greater  is  the  range 
of  motion  of  p  over  the  bridge  wire.  If  the  bridge  wire  is  cali- 
brated (that  is  the  value  of  p  determined)  after  being  shunted,  no 
error  is  introduced  by  such  shunting,  provided  the  resistances  of 
the  leads  n  and  n\  to  the  bridge  wire  do  not  alter  after  the  calibra- 
tion is  made.  This  possible  error  is  avoided  by  making  these  leads 
of  heavy  copper  cable. 

The  bridge  may  be  provided  with  three  bridge  wires  of  different 
resistances,  any  one  of  which  may  be  used  at  will.  Two  shunts 
may  be  provided  for  the  wire  of  lowest  resistance,  and  thus  there 
is  altogether  the  equivalent  of  five  different  bridge  wires.  This 
arrangement  adapts  the  apparatus  to  making  direct  comparisons 
with  a  wide  range  of  resistances. 

The  bridge,  as  mechanically  designed,  often  consists  of  two 
separate  units. 

One  unit,  which  we  may  designate  "  the  coil  holder,"  consists 
of  a  hard-rubber  base  upon  which  are  mounted  massive  copper 
bars  for  holding  and  connecting  the  ratio  coils  and  the  resistance 
standards,  also  the  commutating  device  for  interchanging  the 
standard  and  the  coil  under  comparison. 

The  other  unit,  which  may  be  called  the  "  bridge,"  consists  of  a 
hard- wood  base  upon  which  are  fastened  the  three  slide  wires  and 
three  scales.  A  sliding  contact  maker  is  also  a  part  of  the 
"  bridge." 

The  two  units  are  joined  together  electrically  by  low-resistance 
cables. 

The  Carey-Foster  method,  as  above  described,  is  especially  useful : 

(1)  For  comparing  with  fundamental  standards  of  resistance, 
coils  of  approximately  the  same  resistance.     For  this  purpose  the 
bridge  is  adapted  to  comparing  coils  with  standards  of  resistance. 

(2)  Coils  may  be  compared  with  the  standards  when  they  differ 
in  value  from  the  standards  very  considerably,  provided  one  is 
supplied  with  a  resistance  set  of  only  very  moderate  accuracy. 
If  the  resistance  coil  has  a  value  higher  than  the  standard,  it  is 
shunted  with  the  coils  of  a  resistance  box  until  the  value  of  the 


66  MEASURING  ELECTRICAL  RESISTANCE         [ART.  403 

shunted  combination  is  sufficiently  near  that  of  the  standard  to 
enable  an  exact  balance  to  be  obtained  with  one  of  the  bridge 
wires.  If  the  coil  under  comparison  has  a  lower  value  than  the 
standard,  then  the  standard  is  shunted.  The  exact  method  of 
procedure  will  be  explained  later. 

(3)  A  valuable  application  of  the  method  is  the  obtaining  of 
temperature  coefficients  of  resistance  coils  or  specimens  of  wire 
of  any  kind,  and  exceedingly  accurate  results  can  be  obtained. 

(4)  The  method  is  very  useful  in  adjusting  a  number  of  resist- 
ance spools  to  the  same  value.     For  this  a  bridge  wire  is  used,  of 
the  same  kind  and  size  as  the  wire  of  the  resistance  spools.     A 
single  reading  will  then  give  at  once  without  any  calculation  the 
exact  length  of  wire  that  must  be  cut  off  to  make  the  resistance  of 
the  spool  equal  to  that  of  the  standard. 

(5)  A  Carey-Foster  bridge,  together  with  a  few  reliable  standard 
resistances,  enables  one  to  check  up  the  coils  of  a  Wheatstone 
bridge  or  standard  resistance  box  with  great  accuracy. 

(6)  Very  low  resistances,  such  as  the  contact  resistance  of  plugs 
in  a  resistance  box  or  the  resistance  of  lead  wires,  are  conveniently 
measured  with  the  Carey-Foster  bridge.     For  such  cases  a  thick 
metal  bar  of  known  resistance  may  occupy  the  place  of  a  standard 
resistance  coil. 

When  one  wishes  to  compare  with  a  standard  resistance  another 
resistance  differing  from  it  considerably,  we  may  do  so  by  shunt- 
ing the  standard  resistance,  if  it  has  the  higher  value  or  the 
resistance  to  be  compared,  if  it  has  the  higher  value.  The  pro- 
cedure is  then  the  same  as  in  other  cases. 

Suppose  first  that  the  resistance  under  comparison  is  greater 
than  the  standard.  Then  if  we  shunt  this  resistance  with  a  known 
resistance  a-  we  have 


where  D  =  a2  — 

or  we  obtain  Si  =  a^  .  (13) 

a  —  o  ~r  pU 

If,  second,  the  resistance  is  less  than  the  standard,  the  standard 
may  be  shunted  with  a  resistance  a  and  we  have 

-81  =  Wr-  -  PD.  (14) 


ART.  403]         THE   WHEATSTONE-BRIDGE  NETWORK 


67 


The  value  of  a  does  not  need  to  be  known  with  great  accuracy 
if  its  value  is  considerably  greater  than  the  resistance  which  it 
shunts.  Thus  if  a  =  100  S,  and  is  in  error  one-tenth  of  one  per 
cent,  the  calculated  value  of  S  will  be  in  error  only  about  one- 
hundredth  of  one  per  cent. 

It  may  happen  that  we  wish  to  compare  a  coil  of,  say,  nominally 
10,000  ohms  resistance  with  a  standard  coil  of  that  resistance.  In 
general  the  coil  to  be  compared  will  differ  so  much  from  the  standard 
that  a  balance  cannot  be  obtained  with  the  slider  on  the  bridge 
wire.  In  this  case  a  variable  resistance  may  be  inserted  in  series 
with  the  coil  of  lower  resistance  and  adjusted  until  a  balance  is 
made  possible  with  the  slider  on  the  bridge  wire.  This  added 
resistance,  if  not  known,  can  then  be  measured  and  its  value  added 
or  subtracted  as  may  be  required  from  the  coil  being  compared. 

Again  it  may  be  required  to  measure  a  resistance  which  is  lower 
than  the  total  length  of  one  of  the  two  bridge  wires.  Such  a  case 
would  be  where  we  wish  to  obtain  the  value  of  the  lead  wires 
connecting  the  mercury  cups  of  the  bridge  to  a  coil  in  a  resistance 
box.  This  case  is  .easily  met  by  connecting  together  one  set  of 
mercury  cups  by  a  copper  bar  of  negligible  resistance,  or  a  bar  of 
resistance  metal  of  negligible  temperature  coefficient  and  known 
resistance. 

Such  a  bar,  then,  takes  the  place  of  a  standard  coil,  serving,  in  fact, 
as  a  standard  of  zero  or  very  low  resistance.  If  RQ  is  the  resistance 
of  this  short-circuiting  rod,  and  Rm  the  low  resistance  being  meas- 
ured we  have  the  same  relation  as  that  expressed  by  Eq.  (6),  namely, 
Rm  =  RQ  —  p  (a-2,  —  0,2). 

Following  are  a  few  specimen  readings  which  will  serve  to 
illustrate  a  satisfactory  method  of  procedure.  Some  readings  are 
also  given  below  which  were  taken  in  calibrating  one  of  the  bridge 
wires.  The  numerical  calculation  of  the  final  result  is  also  given. 


a  i 

A] 

«r-«i 

Temp, 
st'd. 

Temp, 
coil,  a/ 

Temp, 
coil,  <Z( 

Mean 
temp. 

Res.,  st'd. 

Res.  of  coil 
at  observed 
temp. 

512.0 

510.5 

+     1.5 

23.8 

19  '.4 

19.6 

19.5 

100.0068 

100.0062 

471.0 

554.0 

-  83.0 

23.8 

26.5 

26.5 

26.5 

100.0068 

100.0403 

453.0 

586.0 

-133.0 

24.6 

30.4 

30.2 

30.3 

100.0070 

100.0607 

Here  column  (1)  gives  the  readings  in  millimeters  of  the  length  of 
scale  to  the  left  of  the  contact  when  an  end  A  of  the  commutator 


68 


MEASURING  ELECTRICAL  RESISTANCE         [ART.  403 


is  towards  the  ratio  coils.  Column  (2)  gives  the  scale  readings 
after  reversing  the  commutator.  Column  (3)  gives  the  difference 
between  the  two  sets  of  readings,  regard  being  had  to  the  sign. 
Column  (4)  is  the  observed  temperature  in  degs.  C.  of  the  standard. 
Column  (5)  is  the  observed  temperature  of  the  coil  at  the  time  of 
taking  reading  a/.  Column  (6)  is  the  temperature  of  the  same 
at  the  time  of  taking  reading  ai.  Column  (7)  is  the  mean  of  the 
two  temperatures.  Column  (8)  is  the  resistance  of  the  standard 
corresponding  to  the  observed  temperatures  in  column  (4),  these 
resistances  being  calculated  using  the  certified  value  and  tempera- 
ture coefficient  of  the  standard,  or  more  conveniently  read  from  a 
curve  plotted  to  show  the  relations  between  resistance  and  tempera- 
ture of  the  standard.  Column  (9)  gives  the  calculated  resistances 
at  the  different  mean  temperatures  of  column  (7)  of  the  coil  under 
comparison.  When  a  sufficient  number  of  readings  are  taken  for 
different  temperatures  a  very  accurate  curve  showing  the  changes 
in  resistance  due  to  temperature  changes  may  be  plotted. 

With  the  particular  bridge  wire  used  in  obtaining  the  above 
results  p  =  0.000404  ohm  per  millimeter  of  wire,  and  thus,  if  S 
be  the  resistance  of  the  standard  given  in  column  (8)  for  particular 
temperatures,  the  results  in  column  (9)  are  obtained  from  the 
relation  Si  =  S  -  0.000404  (a/  -  ai).  See  Eq.  (7),  par.  403.  The 
actual  readings  and  the  resistances  used  for  obtaining  the  above 
value  of  p  are  as  follows: 

10-ohm  standard,  No.  1592,  called  Si.  10  ohms  shunted  with 
1000  ohms,  called  Si'. 

Si'  = j — p  =  9.90300, 


10.00206       1000 
S  -  Si  =  9.90300  -  10.00206  =  -  0.09906. 


«l 

<Z2 

•V 

a2' 

Temp, 
standard 

Res.,  st'd. 

Res.  shunted. 

526.5 
526.5 
526.1 
526.1 

526.3 

497.0 
497.5 
497.5 
497.5 

497.4 

649.0 
648.5 
648.5 
648.5 

648.4 

374.0 
374.5 
374.5 
374.5 

374.4 

30.6 

a 

te 
i  ( 

10.00206 

« 

u 
11 

10.000 

tl 

1C 

(( 

mean  readings. 

p  = 


Si' 


-  0.09906 


-  aif  -  a*      374.4  +  526.3  -  648.4  -  497.4 


ART.  404]          THE  WHEATSTONE-BRIDGE  NETWORK 


69 


or 


P  = 


-  0.09906 


-  0.000404. 


-  245.1 

404.  Galvanometer  Resistance.  Measured,  Using  the  "  Second 
Property  "  of  the  Bridge. —  This  method,  called  Kelvin's  method, 
offers  a  good  example  of  the  employment  of  the  "  second  property  " 
of  the  bridge.  The  method  would  be  employed  where  only  one 
galvanometer,  the  one  whose  resistance  is  to  be  determined,  is 
available.  The  method  is  equally  applicable  for  determining  the 
resistance  of  a  millivoltmeter. 

Make  the  connections  as  shown  in  Fig.  404. 


ri  r2 

-AMM/1 — ' — WVWWWWWWVN 


Ba 


FIG.  404. 


The  resistance  R  should  be  chosen  about  equal  to  the  resistance 
g  of  the  galvanometer.  The  E.M.F.  of  the  battery  Ba  will  gen- 
erally be  too  great  to  be  applied  directly  to  the  bridge,  as  the 
galvanometer  will  deflect  off  its  scale.  The  E.M.F.  may  be  re- 
duced to  -  -  of  its  value  in  the  manner  indicated  in  Fig.  404. 


It  should  be  so  adjusted  that  the  deflection  of  the  galvanometer 
(or  other  deflection  instrument  having  a  resistance  g  to  be  deter- 
mined) remains  upon  the  scale  at  all  times.  K  is  a  key.  It  will 
be  found  that  the  deflection  of  the  galvanometer  is  varied  when 
the  sliding  contact  p  is  moved  along  the  slide  wire  ab  and  the 
deflection  will  also  vary  when  the  key  is  closed,  except  for  one 
position  of  the  slider  p.  The  measurement  is  made  by  finding 
this  position  of  p  where  the  deflection  of  the  galvanometer  remains 
unaltered  when  the  key  K  is  open  or  closed.  When  this  position 


70  MEASURING  ELECTRICAL  RESISTANCE         [ART.  405 

is  found  the  battery  circuit  and  the  key  circuit  are  conjugates, 
and  then  by  the  "  second  property  "  of  the  Wheatstone  bridge 
(§  400)  we  have 

R         c  I  —  c  n 

•jTtt   or  9=—R>  (1) 

when  I  is  the  length  of  the  bridge  wire  and  c  is  the  distance  from 
end  a.  This  method,  due  to  Lord  Kelvin,  gives  very  good  results 
when  correctly  applied.  The  source  of  E.M.F.  applied  to  the  bridge 
and  the  key  may  be  interchanged  when  the  same  results  follow. 

A  trial  was  made  of  the  above  method  in  measuring  the  resist- 
ance of  a  Weston  millivoltmeter.  The  connections  were  made 
as  in  Fig.  404.  A  slide-wire  meter  bridge  was  used,  the  resistance 
of  the  slide  wire  being  0.1397  ohm  per  centimeter  (a  much  higher 
resistance  than  is  usual  in  this  type  of  bridge). 

Ba  was  a  storage  cell.  R  was  selected  10  ohms.  The  value 
found  for  the  resistance  corresponding  to  the  200-millivolt  scale 
was 

I-  c  n      100  -  49.1  ,„ 

g  = R  = 2771 10  =  10.366  +  ohms. 

c  ~ty .  i 

A  second  trial  was  made  in  which  R  was  again  chosen  10  ohms, 
but  use  was  made  of  extension  coils  each  of  100  ohms.  These 
were  inserted  at  the  ends  of  the  bridge  wire,  as  shown  in  Fig.  401a 
above.  Remembering  that  the  resistance  of  the  bridge  wire  had 
been  found  to  be  0.1397  ohm  per  centimeter  the  result  obtained 
becomes 

p      100 +  (100- 36)  X  0.1397,  n     inQ79 

R  -         10Q  +  36XQ.1397  =  10'372  °hms' 

Here  l\  =  the  total  length  of  the  bridge  wire  in  centimeters  mul- 
tiplied by  the  resistance  of  the  wire  per  centimeter,  and  Ci  =  the 
length  in  centimeters  from  left  end  of  wire  of  sliding  contact  mul- 
tiplied by  its  resistance  per  centimeter.  This  last  result  differs 
from  the  first  by  about  0.06  per  cent  which  shows  that  both 
methods  are  fairly  precise. 

The  method  is  too  insensitive,  as  applied  above,  for  measuring 
the  high  resistance  of  a  voltmeter. 

1  405.  Calibration  of  Bridge  Wire.  —  When  a  slide-wire  bridge 
is  used  and  high  precision  is  required  in  resistance  measurements 
it  is  necessary  to  calibrate  the  slide  wire  for  uniformity  of  resist- 
ance. A  manganin  wire,  if  carefully  selected  and  handled,  will 


ART.  405]         THE  WHEATSTONE-BRIDGE  NETWORK 


71 


be  fairly  uniform  in  resistance.  Nevertheless  this  fact  should  be 
determined,  and,  if  it  is  found  not  to  be  uniform,  corrections  should 
be  applied  in  as  simple  a  manner  as  possible.  We  may  make  the 
calibration  and  obtain  an  expression  which  will  give  the  values  of 
the  corrections  to  apply  as  follows: 

In  Fig.  405a,  p,  p,  etc.,  represent  ten  or  more  resistance  coils 
which  are  exactly  alike  in  resistance.     They  may  have,  conven- 


•If 


m        5         6 
H[^ VWWW 


FIG.  405a. 

iently,  a  resistance  of  about  100  ohms  each.  The  absolute  resist- 
ance of  these  coils  does  not  need  to  be  known  and  it  is  easy  to 
adjust  a  number  of  such  coils  to  be  equal  in  resistance  with  the 
aid  of  a  Wheatstone  bridge  of  very  simple  construction.  One  of 
the  terminal  wires  of  each  spool  may  be  shortened  or  lengthened 
by  trial  until  each  coil  matches  some  one  taken  as  a  standard. 
If  the  spools  are  wound  with  manganin  wire  the  difficulty  of 
maintaining  the  temperature  sufficiently  constant  while  the  adjust- 
ments are  being  made  is  not  great. 

The  terminals  of  these  coils,  n  in  number  and  joined  in  series, 
are  connected  by  means  of  heavy  leads  cp  and  dq  to  the  terminals 
of  the  bridge  wire  to  be  calibrated.  It  is  assumed  that  this  wire 
can  be  divided  into  divisions  of  known  length  by  means  of  a  scale 
near  to,  or  lying  underneath,  the  wire.  A  cell  of  battery,  with 
some  resistance  P  in  series,  is  joined  to  the  ends  of  the  bridge 
wire  at  the  points  p  and  q  (not  at  c  and  d).  A  galvanometer  has 
one  terminal  connected  to  a  traveling  contact  and'  the  other 
terminal  t  is  arranged  so  that  connection  may  be  made  at  the 
various  points  between  the  coils,  as  1,  2,  3,  etc. 


72  MEASURING  ELECTRICAL  RESISTANCE         [ART.  405 

Then,  if  the  contact  t  is  at  point  1  a  balance  of  the  galvanometer 
will  be  obtained  when  the  sliding  contact  is  at  a  distance  li  from 
the  end  of  the  wire,  or,  in  general,  when  the  contact  t  is  at  any 
point  m'  between  the  coils,  a  balance  will  be  obtained  when  the 
sliding  contact  is  at  a  distance  lm  from  the  end  p  of  the  wire.  As 
the  coils  p,  p,  etc.,  are  all  of  equal  resistance  it  is  evident  that  the 
wire  becomes  in  this  way  divided  up  into  n  lengths  of  equal  re- 
sistance. If  the  wire  is  not  uniform  in  resistance  these  n  lengths 
will  not  be  equal. 

Let  R  —  the  total  resistance  of  the  wire  from  p  to  q  and 
L  =  the  total  length  of  the  wire, 

then  —  R  will  be  —  of  the  total  resistance. 
n  n 

If  we  call  TFi  the  resistance  of  -  of  the  length  of  the  wire  or,  in 
general,  Wm  the  resistance  of  —  of  the  length  of  the  wire,  we  shall 

71 

have 

Wm=™R  +  8rm,    or     Wm=^R-8rm, 

according  as  Wm  >     or     <  —  R.     Choosing  the  first  case 

8rm=Wm-™R.  (1) 

Here  drm  is  the  small  difference  in  resistance  between  —  of  the 

n 

length  of  the  wire  and  —  of  the  total  resistance  of  the  wire. 

Again,  let  lm  =  the  distance  from  end  p  at  which  a  balance  is 
obtained  when  the  terminal  t  is  between  the  mth  coil  and  the 
(m  +  l)th  coil.  Then 

n  n 

Choosing  the  first  case 

X7  7  ^  T  ffy\ 

Oim  =  lm—  —  Li.  (Z) 

Here  8lm  expresses  the  difference  in  the  lengths  (at  points  1,  2, 
3,  etc.  .  .  .  n)  where  a  balance  comes  on  the  actual  wire  and  where 
it  would  come  if  the  wire  were  uniform. 


ART.  405]         THE  WHEATSTONE-BRIDGE   NETWORK  73 

It  is  also  evident  that  within  an  error  of  the  second  order  which 
need  not  be  considered, 

5rm=  jf-R.  (3) 

The  results  expressed  in  Eq.  (2)  should  be  plotted  in  a  curve  as 
follows: 

Divide  the  axis  of  ordinates  into  distances  li,  1%,  13,  .  .  .  ln, 
corresponding  to  the  various  positions  found  upon  the  wire  where 
a  balance  was  obtained.  At  each  of  these  points  raise  ordinates 


+10 


+  5 


-  5 


-10 


-15 


100          200          300  400  500          600 

Millimeters 

FIG.  405b. 


700 


900        1000 


equal  to  8li,  5?2,  5Z3,  etc.,  to  5Z(n_i).  The  values  of  dlm  may  have 
different  signs  and  some  of  the  ordinates  may  extend  below  the 
axis.  Thru  the  ends  of  the  ordinates  draw  a  smooth  curve. 
From  this  curve  values  which  lie  between  the  determined  values 
of  dlm  may  be  taken. 

Fig.  405b  shows  such  a  curve  drawn  from  actual  observations 
made  upon  a  manganin  wire  1  meter  long.  This  wire  had  been 
scraped  in  its  middle  portion  for  about  one  third  of  its  length  in 
order  to  exaggerate  its  inequality. 

We  can  now  find  an  exact  expression  for  the  value  of  any  un- 
known resistance  when  measured  with  a  slide-wire  bridge  in  which 
the  calibrated  wire  is  used. 


74 


MEASURING  ELECTRICAL  RESISTANCE         [ART.  405 


In  measuring  the  resistance  X  with  the  slide-wire  bridge  in  the 
manner  indicated  in  Fig.  405c,  we  have 

W 

X  =  m     r  (^ 

~  R-Wmr' 

where  R  is  the  total  resistance  of  the  wire  and  r  the  known  resist- 
ance. 


P      R-Wm 
FIG.  405c. 


FromEq.  (1) 
Hence 


X  = 


mR  -f  ndr 


r. 


nR  —  (mR  +  ndrm) 
Placing  in  Eq.  (5)  the  value  of  drm  given  in  Eq.  (3)  we  have 


m ---- 


X  = 


r. 


(5) 


(6) 


Since  n  and  L  occur  as  a  ratio  we  can  express  them  in  any  units 
we  choose  provided  the  same  units  are  used  for  each.  Take 
L  =  1000,  corresponding  to  1000  millimeters  for  a  slide  wire 
1  meter  long,  and  take  n  equal  1000.  Then 

8lm 


X  = 


m 


1000  -  (m  +  dlm) 


r. 


(7) 


To  use  Eq.  (7)  proceed  as  follows: 

Obtain  a  balance  of  the  bridge  by  sliding  the  contact  p  along  the 
wire.  Note  the  distance  in  millimeters  (or  in  divisions  equal  to 
TuW  °f  the  length  of  the  wire,  if  ( this  is  not  1  meter  long)  from 
the  left-hand  end  of  the  wire  to  the  point  of  balance.  This  dis- 
tance will  be  m.  From  the  curve  find  the  value  of  dlm  which 
corresponds  to  m  scale  divisions  (to  millimeters  in  the  case  of  a 


ART.  406]         THE  WHEATSTONE-BRIDGE  NETWORK 


75 


wire  1  meter  long).     Add  this  value  to  the  reading  m,  and  call  the 
result  a.     Then 

a 


X 


1000  -  a 


r. 


(8) 


The  quantity 


may  be  taken  from  the  table  (Appendix 


1000  -  a 

I,  1)  as  in  other  cases.  If  a  calibration  of  a  slide  wire  is  made 
carefully  in  this  way  and  used,  a  slide-wire  bridge  may  become 
an  accurate  device  for  measuring  resistance. 

A  trial  was  made  of  the  application  of  this  correction.  The 
wire  used  was  the  same  one  for  which  the  calibration  curve  is 
shown  plotted  in  Fig.  405b.  The  known  resistance  r  was  given 
different  values  and  a  resistance  was  measured  using  the  con- 
nections shown  in  Fig.  405c.  The  values  obtained  for  X  were 
calculated  by  Eq.  (7)  and  are  exhibited  below. 


r  in 

m  in 

x, 

x. 

Per  cent 

ohms 

millimeters 

MM 

m  +  6Zw 

measured 

true  value 

error 

25 

790  8 

+    8.95 

799.75 

99.84 

100 

-0  16 

100 

507  1 

-    7.00 

500.10 

100.04 

100 

+  0  04 

200 

348.2 

-  14.80 

333.40 

100.03 

100 

+  0  03 

300 

261  5 

-11.70 

249.80 

99.89 

100 

-0  11 

The  failure  to  get  higher  precision  resulted  from  the  crudeness 
of  the  apparatus.  This  consisted  of  a  wire  stretched  upon  a 
wooden  meter  stick  with  no  provision  for  the  sliding  contact  other 
than  the  blade  of  a  knife  held  in  the  hand. 

406.  The  "  Kelvin- Varley  Slides." — This  is  a  device  whereby 
the  slide  wire  of  a  slide-wire  bridge  may  be  replaced  with  sets  of 
resistance  coils.  The  latter  are  so  disposed  that  without  an 


excessive  number  of  coils  the  ratio 


may  be  obtained 


1000  -  a 

and  a  be  varied  in  steps  as  small  as  desired.  This  device  con- 
nected in  a  Wheatstone-bridge  network  is  represented  in  Fig.  406. 
In  the  arrangement  shown  there  are  in  row  1,  11  coils,  each  of 
resistance  r.  Spanning  always  two  coils  are  the  two  contacts 
PI,  pi,  which  may  be  moved  together  over  the  studs  or  blocks  to 
which  the  coil  terminals  are  joined.  In  row  2  there  are  also 

11  .coils,  each  of  resistance  - ,  and  two  traveling  contacts  p%,  p2) 

o 


76 


MEASURING  ELECTRICAL  RESISTANCE         [ART.  406 


which  always  span  two  coils  of  this  row.     In  row  3  there  are 

10  coils,  each  of  resistance  ^=  and  one  traveling  contact  p. 

&o 

Now,  it  is  evident,  if  the  resistance  from  contact  pi  to  pi  of 
all  below  these  contacts  is  equal  to  2  r,  that  two  coils  of  row  1 
are  always  shunted  with  a  resistance  equal  to  these  two  coils. 
Thus  the  resistance  from  pi  to  p\,  when  the  contacts  bear  upon 


the  studs,  will  be  r.  There  will  be  thus  in  row  1  ten  equal 
resistances,  each  of  value  r.  Similarly,  if  the  resistance  from 

contact  pz  to  pz  of  all  below  these  contacts  is  — ,  then  there  will 

5 

be  ten  resistances  in  row  2,  each  equal  to  -•     Lastly,  in  order  that 

o 

the  total  resistance  of  the  10  coils  of  row  3  shall  equal  —  ,  each 

o 

of  the  10  coils  must  have  a  resistance  ~r-     With  this  arrangement 

the  total  resistance  from  c  to  d  will  be  10  r  ohms.  The  value 
of  the  reading  a  will  now  be  given  by  the  positions  occupied  by 
the  left-hand  contacts.  With  the  positions  shown  in  the  figure 

/i 

becomes 


the  reading  is  a  =  375  and  the  ratio 
375 


1000  -  375 


1000  -  a 
0.6000   (see  table,  Appendix  I,  1). 


The  same  principle  of  subdivision  may  be  indefinitely  extended 
to  read  in  steps  as  small  as  desired.     The  one  represented  in  the 


ART  406]         THE  WHEATSTONE-BRIDGE  NETWORK  77 

diagram  reads  to  1  part  in  1000.  The  Kelvin- Varley  slides  is  a 
device  equivalent  to  a  long  slide  wire.  It  may  be  given  any 
accuracy  of  adjustment  desired.  The  cost  of  the  mechanical 
construction  required  and  the  many  coils  which  must  be  adjusted 
have  limited  the  use  of  this  otherwise  excellent  arrangement. 


CHAPTER  V. 

WHEATSTONE-BRIDGE  METHODS.     VARIABLE  RHEO- 
STAT.    ARRANGEMENTS    OF    RESISTANCES. 
PER     CENT     BRIDGE.      SUGGESTIONS 
•  FOR    USING    BRIDGE. 

500.   Wheatstone-bridge  Methods  with  Variable  Rheostat.  - 

To  carry  out  these  methods,  arrangements  are  provided  for 
setting  the  ratio  arms  to  a  chosen  fixed  ratio,  and  for  varying  the 
resistance  in  the  rheostat  arm  until  a  balance  is  obtained. 


To  change  the  setting  of  the  ratio  and  to  change  the  resistance 
in  the  rheostat  involves,  by  all  ordinary  methods,  contact  resist- 
ances in  at  least  two  arms  of  the  bridge.  This  is  a  possible 
source  of  error  which  will  be  better  understood  from  a  consider- 
ation of  Fig.  500. 

In  this  figure  let  a  and  b  represent  the  two  ratio  arms,  R  the 
rheostat  and  X  the  resistance  to  be  determined.  To  change  the 
ratio  a  to  6  (in  bridges  of  ordinary  construction)  it  is  necessary 
to  change  the  value  of  at  least  one  of  these  resistances.  This 
may  be  accomplished  by  moving  a  point  of  contact,  as  p&,  to 
different  positions  along  the  arm  2  to  3.  But  there  will  be  some 
contact  resistance,  however  good  the  construction,  at  the  point  p^ 

78 


ART.  501]  WHEATSTONE-BRIDGE   METHODS  79 

Call  this  contact  resistance  8b.  Likewise  to  change  the  value 
of  the  rheostat  arm  1  to  4  will  require  with  any  type  of  construc- 
tion, at  least  one  movable  contact,  as  pR,  which  will  have  some 
contact  resistance  which  we  may  call  5R. 

The  equation  of  balance  of  the  bridge,  then,  becomes 

a  R  +  dR 


5b 


If  we  expand  the  second  member  of  Eq.  (1)  and  neglect  the  prod- 
uct 5b  dR  we  find 

(2) 


The  last  term  of  Eq.  (2)  represents  a  necessary  error  in  estimating 
the  value  of  X,  which  is  due  solely  to  contact  resistances  in  two 
arms  of  the  bridge.  In  the  special  type  of  construction  described 
in  par.  510,  the  contact  resistance  66  is  done  away  with  by 
shifting  the  galvanometer  terminal  to  different  positions  along  a 
or  b,  these  being  soldered  together  at  joint  2.  The  error  which 
still  remains  is  the  second  term  of  the  right-hand  member  of  the 
relation 

X  =  -R  +  -dR.  (3) 

a          a 

This  source  of  error  should  be  reduced  to  a  minimum  by  a 
proper  mechanical  construction  of  the  rheostat.  To  accomplish 
this  and  at  the  same  time  make  a  rheostat  which  may  be  con- 
veniently varied  in  small  steps  thru  a  wide  range  has  given  oppor- 
tunity for  a  wide  variety  in  design.  The  resistance  coils  or  units 
may  be  arranged  in  numerous  ways  and  the  mechanical  construc- 
tion for  putting  various  resistance  values  in  circuit  may  be  greatly 
varied.  Both  brass  blocks,  to  be  connected  by  plugs,  and  brushes 
which  can  be  moved  over  studs  arranged  in  the  arc  of  a  circle, 
called  a  dial,  are  extensively  used.  We  shall  now  describe  the 
various  methods  of  arranging  the  coils,  but  will  omit  a  description 
of  mechanical  constructions. 

501.  Arrangements  of  Resistances  in  Wheatstone-bridge 
Rheostats.  —  This  subject  was  discussed  by  the  author  in  an 
article  published  in  the  Electrical  Review,  June  and  July,  1903, 


80  MEASURING  ELECTRICAL  RESISTANCE         [ART.  501 

and  much  of  what  follows  under  this  heading  and  the  next  is 
take  from  that  publication: 

The  fundamental  purpose  of  a  Wheatstone-bridge  rheostat, 
whether  employing  plugs  and  blocks  or  dials  and  sliding  contacts, 
is  to  provide  means  of  obtaining  the  largest  possible  number  of 
values  from  the  fewest  possible  number  of  accurately  adjusted 
resistance  units  and  to  do  this  without  introducing  into  the  circuit 
objectionable  contact  resistances. 

Where  the  number  of  coils  is  made  greater  than  the  least  number 
required  theoretically,  it  is  done  to  give  some  convenience  of 
working  or  to  increase  the  ease  or  simplicity  with  which  the  values 
are  added  up. 

The  theoretical  combinations  of  resistance  coils  .which  are 
possible  are  given  by  the  following  relations:  If  there  are  n  coils 
these  can  be  joined  in  various  combinations,  in  series,  or  in  parallel, 
or  in  parallel  and  series  or  again  in  mixed  arrangements.  The  total 
number  of  combinations  that  can  be  formed,  using  the  coils  singly 
and  by  joining  them  in  series  combinations,  is 

T  =  2n  -  1.  (1) 

The  same  total  number  of  combinations  can  be  formed  if  the  coils 
are  taken  singly  and  joined  in  all  parallel  combinations^  but  since 
the  coils  used  singly  are  the  same  for  both  arrangements,  we 
have  as  the  total  number  of  combinations  possible  for  n  coils  used 
singly  and  joined  in  series  and  in  parallel  combinations 

N  =  2n-l-n+2n-l     or    N  =  2n+1  -  (n  +  2).        (2> 

We  will  not,  in  general,  by  combining  coils  in  all  these  ways,  obtain 
as  many  different  values  of  resistance  as  there  are  combinations 
of  coils,  for  many  of  the  combinations  of  coils  give  the  same  re- 
sistance values  even  tho  the  resistance  of  each  coil  be  different. 
Thus,  the  much-used  set  of  coils,  of  1,  2,  3,  4  ohms,  respectively, 
can  be  joined  in  series  in77  =  24—  1  =  15  different  ways,  but 
in  5  cases  the  same  values  of  resistance  are  repeated,  giving  but 
10  different  values  of  resistance  for  this  series  of  coils.  The  total 
number  of  different  values  of  resistance  which  can  be  obtained 
from  a  given  number  of  coils  becomes  in  any  particular  case  a 
complex  problem,  if  the  number  of  coils  be  large. 

If  coils  are  arranged  in  all  possible  ways,  we  can  get  from  2  coils 
4  arrangements,  from  3  coils  17  arrangements  and  from  4  coils 
106  arrangements.  Thus  it  is  often  possible,  when  only  a  few 


ART.  502]  WHEATSTONE-BRIDGE   METHODS  81 

standards  of  resistance  are  at  hand,  to  obtain,  nevertheless,  a 
large  variety  of  values. 

502.  Rheostat  Coils;  Classical  Arrangements.  —  To  arrange 
resistances  in  combinations  so  a  series  of  values  can  be  quickly 
and  easily  added  up  has  led  to  the  employment  of  at  least  five 
methods,  tho  only  three  of  these  are  now  in  common  use. 

Siemens'  Plan.  —  The  first  plan  is  due  to  Siemens  and  consists 
in  joining  a  series  of  coils  between  blocks,  the  coils  having  the 
values  1,  2,  4,  8,  16,  32,  etc. 

Resistance  is  thrown  in  circuit  by  removing  plugs  from  between 
the  blocks.  This  plan  employs  the  smallest  possible  number  of 
coils  for  attaining  a  given  range  of  values,  but,  as  the  summing 
up  of  the  values  is  not  an  easy  matter,  this  method  is  now 
obsolete. 

The  1,  2,  3,  4  Plan.  —  By  this  plan  all  values  from  1  to  10  ohms, 
in  steps  of  1  ohm,  are  obtained  by  the  coils  1,  2,  3,  4;  all  values 
from  10  to  100  ohms,  in  steps  of  10  ohms,  by  the  coils  10,  20, 
30,  40;  all  values  from  100  to  1000  ohms  in  steps  of  100  ohms,  by 
the  coils  100,  200,  300,  400.  By  this  arrangement  of  coils  there 
must  be  as  many  plugs  as  coils,  the  required  values  being  obtained 
by  withdrawing  plugs.  As  many  plugs  will  be  withdrawn  as  there 
are  coils  in  the  circuit. 

The  1,  2,  2 ,  5  Plan.  —  This  arrangement  of  coils  is  used  in  pre- 
cisely the  same  way  as  the  one  above.  Most  resistance  boxes  in 
which  plugs  are  removed  to  throw  resistance  in  the  circuit  are 
based  on  one  or  other  of  these  two  plans. 

The  1,1,3,5  Plan.  —  1, 1,  3,  5;  10,  10,  30,  50;  etc.,  will  give  ten 
consecutive  values  in  each  unit's  place,  like  the  two  above.  This 
arrangement,  so  far  as  the  author  knows,  has  not  been  used  in  any 
resistance  boxes  placed  upon  the  market. 

There  are  no  other  four  numbers  which  add  up  to  ten  which 
will  give  ten  consecutive  values. 

The  Decade  Plan.  —  In  this  arrangement,  as  ordinarily  applied, 
there  are  9  or  10  one-ohm  coils  for  the  units'  place,  9  or  10  ten-ohm 
coils  for  the  tens'  place,  9  or  10  one-hundred-ohm  coils  for  the 
hundreds'  place,  and  so  on.  Each  series  of  coils  of  the  same  value 
is  designated  a  decade.  The  connections  as  usually  made  are  as 
shown  in  Fig.  502. 

It  is  apparent  from  this  diagram  that  any  value  in  any  one 
decade  can  be  obtained  by  inserting  between  a  bar  and  a  block 


82 


MEASURING  ELECTRICAL  RESISTANCE         [ART.  503 


one,  and  only  one,  plug.  It  also  appears  that  if  several  decades 
are  in  series  any  value  up  to  the  limit  of  the  set  can  be  read  off 
directly  from  the  position  of  the  plugs,  without  any  addition 
whatever. 


10      10       10      10       10      10      .10 


It  is  evident  that  in  the  first  four  plans  the  values  of  resistance 
required  are  obtained  by  withdrawing  plugs  which  must  be  laid 
aside,  with  liability  of  being  lost,  and  one  must  make  sure  that 
all  the  remaining  plugs  are  well  seated  so  as  not  to  introduce 
unknown  contact  resistances.  Furthermore,  as  many  plugs  must 
be  employed  as  there  are  resistance  units  and  to  obtain  any  given 
value  may  require  the  manipulation  of  a  large  number  of  plugs. 
In  the  decade  plan,  upon  the  other  hand,  there  is  but  one  plug 
used  to  a  decade  and  this  is  always  in  service  and  hence  not  readily 
mislaid.  The  use  of  only  one  plug  to  the  decade  makes  it  easy 
to  ascertain  that  this  is  tightly  fitted  in  its  place.  When  the  value 
is  finally  obtained,  by  manipulating  only  the  one  plug  to  the  decade, 
this  value  is  readily  read  off  without  any  mental  summing  up  of 
values.  Again  the  decade  plan  alone  permits  of  obtaining  a  suc- 
cession of  values  by  means  of  sliding  contacts  or  dial  switches,  a 
method  which  is  becoming  deservedly  more  appreciated. 

These  and  other  obvious  advantages  of  the  decade  plan  are  in 
part  offset  by  the  necessity,  if  the  decades  are  arranged  as  in 
Fig.  502,  of  using  a  larger  number  of  resistance  units  than  is 
required  by  the  previous  plans. 

503.  Northrup's  Four-coil  Arrangement.  —  An  arrangement  of 
coils  has  been  devised  by  the  author  which  secures  to  the  decade 


ART.  503] 


WHEATSTONE-BRIDGE  METHODS 


83 


plan  the  further  advantage  of  requiring  no  more  coils  than  the 
1,  2,  3,  4  or  the  1,  2,  2,  5  plans.  This  arrangement  may  be  ex- 
plained as  follows:  Let  the  terminals  of  the  1  ohm  and  2  ohm  coils, 
and  the  points  of  union  of  the  other  coils  be  numbered  (1),  (2), 
(3),  (4).  (5),  as  shown  in  Fig.  503a.  The  current  enters  at  point 
(1)  and  leaves  the  coils  at  point  (5),  traversing  1,  3',  3,  2  =  9 
ohms  in  all.  If  this  series  is  multiplied  by  any  factor  n,  then 
n  (1  +  3'  +  3  +  2)  =  n  9  ohms.  It  will  be  seen  that  if  the 


1  (1) 

AAAVVWWW + 


(2)' 


AA/WWWWV 


AAAAA/WWW 


(3) 


(4) 


A/WWWWW - 

(5) 


FIG.  503a. 


points  (1)  and  (5)  are  connected,  all  the  coils  are  short-circuited, 
and  the  current  will  traverse  zero  resistance.  If  the  points  (2) 
and  (5)  are  connected,  the  3',  3  and  2  ohm  coils  will  be  short- 
circuited  and  the  current  will  traverse  1  ohm.  By  extending  the 
process  so  that  we  connect  two,  and  only  two,  points  at  a  time, 
it  is  possible  to  obtain  the  regular  succession  of  values  n  (0,  1,  2, 
3,  4,  5,  6,  7,  8,  9),  the  last  value  being  obtained  when  no  points 
are  connected.  The  following  table  shows  the  points  which  must 
be  connected  to  obtain  each  of  the  above  values  and  the  coils 
which  will  be  in  circuit  for  giving  each  value: 


84 


MEASURING  ELKCTRICAL  RESISTANCE        [ART.  503 


Value 

Points  connected 

Coils  used 

0 

(5-1) 

0 

1 

(2-5) 

1 

2 

(4-1) 

2 

3 

(2-4) 

1,    2 

4 

(3-5) 

1,    3' 

5 

d-3) 

•3,    2 

6 

(2-3) 

1,    3,    2 

7 

(5-4) 

1,    3,    3' 

8 

d-2) 

3',  3,    2 

9 

(0) 

1,    3',  3,  2 

Fig.  503b  shows  the  method  of  connecting  these  points  two  at 
a  time,  with  the  use  of  a  single  plug. 

The  circles  in  the  diagram  represent  two  rows  of  ten  brass 
blocks  each.  To  the  first  two  blocks  at  the  top  of  the  rows,  the 
points  5  and  1  are  connected,  to  the  second  two  the  points  2  and 
5  are  connected,  and  so  on,  no  points  being  connected  at  the  last 
pair  of  blocks.  It  is  evident  that  if  a  plug  be  inserted  between 
the  blocks  1  and  5,  the  points  1  and  5  are  connected,  giving  the 
value  0;  if  between  the  blocks  2  and  5,  the  points  2  and  5  are  con- 
nected, giving  the  value  1,  and  so  on.  The  value  9  is  obtained 
when  the  plug  is  disposed  of  by  being  inserted  in  the  last  pair  of 
blocks  which  have  no  connections. 

The  only  combinations  of  four  coils  which  will  give  the  dec- 
ade in  the  above  manner  are  the  coils  n  (1,  3,  2',  2),  n  (1,  1',  4,  3) 
and  n  (1,  3,  3',  2)  where  n  may  have  any  value. 

The  above  method  of  obtaining  the  decade  with  only  four  coils, 
as  well  as  the  ordinary  decade  arrangement,  can  be  applied  to  a 
dial  or  sliding  brush  construction.  To  accomplish  this,  20  studs, 
or  ten  pairs,  are  arranged  in  a  circle.  To  make  the  required  con- 
nections, the  pairs  of  studs  at  opposite  ends  of  a  diameter  of  the 
circle  must  be  successively  joined  together.  This  can  be  done 
by  rotating,  with  a  handle  at  the  center  of  the  circle,  a  single 
connecting  bar  or  brush  which  will  join  successively  pairs  of  studs 
to  give  the  values  0,1,  2,  3,  4,  5,  6,  7,  8,  9.  The  20  studs  are  fas- 
tened to  the  top  surface  of  a  hard  rubber  plate  and  there  extends 
down  from  each  stud  thru  the  plate  a  shaft  slightly  longer  than 
a  resistance-unit  or  spool.  On  four  of  these  shafts  (beneath  the 
rubber  plate)  four  resistance  spools  are  mounted  —  the  resistance 
values  of  these  spools  for  a  units'  dial  being  I,  2,  3,  3'  ohms;  for  a 


ART.  503] 


WHEATSTONE-BRIDGE   METHODS 


85 


tens'  dial  10,  20,  30,  30'  ohms;  for  a  hundreds'  dial  100,  200,  300, 
300'  ohms,  etc.  The  ends  of  the  wire  of  the  spools  are  joined, 
one  to  the  end  of  the  shaft  on  which  a  spool  is  mounted,  and 
one  to  the  end  of  an  adjacent  shaft.  The  ends  of  other  shafts 
are  cross-connected  in  the  manner  shown  in  Fig.  503c.  This 
gives  a  view  of  the  connections  as  seen  from  underneath  the 


A 

1,  10,  100  or 
1000  Ohms 


2,  20,  200  or 
2000  Ohms 


3,  30,  300  or 
3000  Ohms 


3,  30,  300  or 
3000  Ohms 


FIG.  503c. 


rubber  plate.  The  circles  in  full  line  represent  ends  of  resistance 
spools.  The  circles  in  dotted  line  represent  the  brass  studs 
upon  the  upper  side  of  the  rubber  plate.  The  larger  circle  in 
dotted  line  represents  a  handle  at  the  center  of  the  circle  of  studs 
located  above  the  upper  side  of  the  plate.  The  bar  or  brush 
shown  in  dotted  line  connects  one  pair  of  studs  after  another  as 
the  handle  is  turned.  The  resistance  value  between  the  points 
A  and  B  with  the  bar  in  the  position  shown  in  Fig.  503c  is  5  ohms. 
Among  other  advantages  of  this  method  of  arranging  four 
resistance  units  to  give  the  decade  in  dial  form  may  be  men- 
tioned the  following:  The  traveling  bar  or  brush  contact  can  be 
rotated  continuously  in  either  direction,  as  there  are  no  electric 
connections  made  to  it.  Hence,  one  can  pass  directly  from  the 
value  0  to  9  by  turning  back  one  stud.  The  construction  is  very 
economical  and  there  are  only  four  coils  in  place  of  nine  to  put 
and  keep  in  accurate  adjustment.  It  is  easy  to  construct  the 
traveling  bar  of  a  number  of  thin  copper  leaves,  which  make  a 
right  angle  turn  at  each  end,  so  as  to  give  an  end  bearing  of  many 
copper  leaves  upon  the  faces  of  the  brass  studs,  thus  securing  a 
certain  and  low-resistance  contact  with  the  studs.  The  disad- 
vantages are  slight,  one  being  that  20  instead  of  10  studs,  as  in 


86  MEASURING  ELECTRICAL  RESISTANCE         [ ART.  504 

the  ordinary  arrangement  of  coils,  are  required.  Also,  at  those 
times  where  the  contact,  in  passing  from  one  stud  to  the  next, 
joins  both  together,  the  resistance  value  is  not  that  of  the  stud 
touched  which  reads  the  lower  value,  but  is  some  odd  value  of 
resistance.  Hence,  in  using  the  dials  the  battery  or  galvanometer 
key  should  be  open  while  the  dial  is  being  turned.  This  objec- 
tion has  no  weight  in  Wheatstone-bridge  work,  but  it  has  some 
weight  in  certain  classes  of  work  where  it  is  desirable  to  follow 
rapidly  changing  resistances  with  the  battery  and  galvanometer 
keys  closed. 

On  the  whole  this  type  of  dial  construction,  used  in  an  ordinary 
Wheatstone  bridge,  is  economical,  accurate,  and  highly  satis- 
factory in  service. 

504.   Five-coil  Combinations.  —  The  author  pointed  out  *  that 
by  using  five  coils  the  decade,  0  to  9  inclusive,  can  be  obtained  in 
a  manner  similar  to  that  used  for  obtaining  the  decade  with  four 
coils,  as  described  above,  from  the  values, 
n(l,  1,2,2,3), 
n  (1,2,  2,2,2), 
and  n(l,  1,  1,  1,5). 

Also  that  eleven  values,  namely  0  to  10  inclusive,  may  be  obtained 
from  the  following  arrangements  of  five  coils: 

n  (1,2,  3,  2,2), 

*  (1,5,1,  1,2), 

n(l,  3,  1,3,  2), 

n(l,  1,  1,3,4), 

n(l,  1,  1,4,  3), 

n(l,  1,4,  1,3), 

n  (1,3,  1,3,2), 

n(l,2,  1,4,2). 

It  was  further  pointed  out  that  by  traveling  a  single  contact, 
in  the  manner  used  with  the  four-coil  arrangement,  the  general 
method  may  be  indefinitely  extended.  Any  number  of  successive 
values  may  be  obtained  with  the  use  of  many  less  coils  than  the 
values  obtainable.  Thus  from  the  seven  coils, 

n(l,  1,3,1,3,3,2), 

we  can  get  fifteen  consecutive  values  inclusive  of  0.     It  was  also 
*  Electrical  Review,  July  18,  1903,  Vol.  43,  page  75. 


ART.  507] 


WHEATSTONE-BRIDGE   METHODS 


87 


pointed  out  that  in  the  above  methods  a  sliding  contact  or  a  dial 

switch  may  be  used  in  place  of  a  plug  for  making  the  connections. 

505.   Decade  System  of  Feussner.  —  In  Fig.  505  is  shown  a 


disposition  (credited  to  M.  Feussner)  of  blocks  and  resistance 

units,  whereby  the  values  0,  1,  2,  3,  4,  5,  6,  7,  8,  9  may  be  obtained 

by  using  five  coils  and  only  one 

traveling    plug.       The    resistance 

units    used    are   n  (1,  1,   1,  1,  5) 

where   n    has    values    1,  10,   100, 

1000,  etc.      An  examination  of  the 

figure  makes  the  system  easy  to 

understand.      The  four-coil  decade 

being  still   better,  this  method  is 

not  likely  to  be  much  used  in  this 

country. 

506.  Decade  System  of  Irving 
Smith.  —  In  Fig.  506  is  shown  a 
disposition  of  5  coils  consisting  of 
n  (1,  2,  2,  2,  2)  where  n  has 
values  1,  10,  100,  1000,  etc., 
whereby  the  decade  may  be  ob- 
tained by  traveling  a  single  plug. 
The  system  is  easily  understood 
from  an  examination  of  the  figure. 


FIG.  506. 


Mr.  Smith  has  used  this  system  with  a  dial  construction  to  which 
it  is  well  adapted. 

507.    Multiple  Arrangements.  —  If  it  be  required  to  obtain  a 
very  low  contact  resistance  of  the  plugs,  to  enable  the  last  decade 


88  MEASURING  ELECTRICAL  RESISTANCE         [ART.  507 

row  to  be  varied  with  some  precision  in  steps  of  0.01  ohm  or  even 
0.001  ohm,  then  a  multiple  arrangement  of  coils  may  be  employed 
with  advantage.  The  principle  whereby  regularly  increasing 
values  of  resistance  are  obtained  by  joining  coils  in  multiple  is 
given  in  Fig.  507. 


FIG.  507. 

It  may  easily  be  shown  that  when  ten  coils  have  values  n  (2,  6, 
12,  20,  30,  42,  56,  72,  90,  10)  ohms,  where  n  has  any  value,  their 
resistance  when  all  are  joined  in  multiple  is  n  ohms.  Also  the 
nine  coils  n  (6,  12,  20,  30,  42,  56,  72,  90,  10)  have  the  resistance 

2  n  ohms  when  joined  in  multiple,  the  eight  coils  n  (12,  20,  30,  42, 
56,  72,  90,  10)  the  resistance  3  n  ohms,  the  seven  coils  n  (20,  30, 
42,  56,  72,  90,  10)  the  resistance  4  n  ohms,  etc.,  to  n  (10)  which 
has  the  resistance  10  n  ohms.     Thus,  with  ten  coils  arranged  as 
in  the  figure  and  having  the  above  values,  all  values  in  steps  of 
n  (I)  ohms  from  0  to  10  n  ohms  may  be  obtained.    Thus,  referring 
to  the  figure,  if  the  plugs  0,  1,  2,  3,  .  .  .  10  are  all  in,  the  resist- 
ance from  a  to  b  is  0  ohm,  if  1,  2,  3,  .  .  .  10  are  all  in,  the  resist- 
ance is  1  ohm.     If  plugs  0,  1,  2  are  removed,  the  resistance  is 

3  ohms,  etc.,  that  is,  the  resistance  obtained  between  a  and  b  will 
always  be  that  which  is  stamped  opposite  the  lasb  plug  toward 
the  left,  which  is  not  removed.     Since  the  current  passes  thru  all 
the  plugs  not  removed,  in  parallel,  the  contact  resistance  is  greatly 
reduced  and  this  contact  resistance  decreases  as  the  resistance 
value  plugged  decreases. 

This  method  is  an  excellent  arrangement  when  it  is  required 
to  obtain  ten  regularly  ascending  values  of  very  low  resistance, 
as,  for  example,  0.001,  0.002,  0.003  .  .  .  0.01  ohm.  Greater  pre- 
cision of  adjustment  can  be  gotten  and  maintained  by  this  parallel 
arrangement  of  coils,  of  relatively  high  resistances,  than  from  coils 
arranged  in  series. 

If  the  entire  rheostat  is  based  upon  this  principle  and  has  several 


ART.  508] 


WHEATSTONE-BRIDGE   METHODS 


89 


decades  it  would  only  be  required  to  have  the  values  run  from 
0  to  9  n  ohms  in  the  decades,  except  the  one  of  lowest  denomina- 
tion. For  obtaining  the  succession  of  values  from  0  to  9  n  ohms 
the  coils  so  joined  in  parallel  are  n  (2,  6,  12,  20,  30,  42,  56, 
72,  9)  ohms.  This  decade  plan  is  deserving  of  more  attention 
than  it  has  received,  for  excellent  results  may  be  obtained  for 
accurately  varying  resistance  in  steps  as  low  as  0.001  ohm. 

The  above  described  selections  and  dispositions  of  resistance 
units  for  Wheatstone-bridge  rheostats  comprehend  those  known 
which  are  of  interest.  We  shall  describe  now  the  methods  in  use 
for  arranging  the  resistances  in  the  ratio  arms  of  a  bridge. 

508.  Arrangements  of  Resistances  for  the  Ratio  Arms  of 
Wheatstone  Bridges.  —  The  most  simple  disposition  of  ratio 


1000       100 


10 


1000 


<D 


Ga 


FIG.  508a. 

coils,  and  one  which  is  found  in  most  inexpensive  Wheatstone- 
bridge  boxes,  made  for  laboratories  and,  the  use  of  students,  is 
shown  in  Fig.  508a.  The  required  ratio,  as  10  to  100,  is  obtained 
by  withdrawing  a  plug  from  each  of  the  arms  A  and  B.  The  con- 
tact resistance  of  the  remaining  plugs  in  each  arm  enters  into  the 
ratio  and  there  is  no  provision  made  for  reversing  the  two  arms. 

In  Fig.  508b  is  shown  how  connections  and  plugs  may  be  dis- 
posed so  that  the  ratio  arms  may  be  reversed. 

This  is  the  classical  arrangement  which  is  used  in  the  so-called 
"  English  Post  Office  "  bridge.  By  changing  two  plugs  from 
holes  1,  2,  to  holes  3,  4,  the  positive  pole  of  the  battery  is  joined 
to  the  end  of  A  and  the  negative  pole  to  the  end  of  B  or  vice 


90 


MEASURING  ELECTRICAL  RESISTANCE         [ART.  508 


versa.  By  so  reversing  the  ratio  arms  and  twice  balancing  the 
bridge  and  then  taking  the  mean  value  of  X  to  be  the  true  value, 
any  errors  of  adjustment  in  the  ratio  coils  is  eliminated  when 
unity  ratio  is  used.  However,  it  is  always  desirable  so  to  choose 
the  values  in  the  ratio  arms  that  a  large  portion  of  the  rheostat 
will  be  required  for  getting  a  balance,  and  it  is  only  when  the 
ratio  is  unity  that  this  can  be  done  with  the  ratio  coils  used  both 
direct  and  reversed.  This  requirement  limits  the  usefulness  of 
reversible  ratio  arms  to  those  cases  where  unity  ratio  may  be 
used.  Thus,  suppose  the  capacity  of  the  rheostat  is  10,000  ohms, 
and  may  be  varied  in  steps  of  1  ohm.  To  set  the  bridge  to  an 
accuracy  of  0.1  of  1  per  cent  would  require  that  at  least  1000  ohms 


FIG.  508b. 

of  the  rheostat  be  brought  into  service.  If  a  resistance  of  900 
ohms  is  to  be  measured,  a  ratio  of  10  to  1,  with  the  ratio  arms 
direct,  would  utilize  9000  ohms  of  the  rheostat,  but  when  the  ratio 
arms  are  reversed  only  90  ohms  of  the  rheostat  .could  be  utilized, 
which  would  necessitate  a  setting  that  might  be  less  accurate  than 
0.5  of  1  per  cent.  In  such  a  case  one  would  either  have  to  aban- 
don the  advantage  of  reversible  ratio  arms  and  set  the  ratio  10 
to  1,  or  use  unity  ratio  and  reverse  the  ratio  arms,  in  which  case 
the  rheostat  setting  would  be  900  ohms,  which,  being  variable 
in  steps  of  1  ohm,  would  permit  settings  to  1  part  'in  900.  The 
advantage,  then,  of  reversible  ratio  arms  is  chiefly  confined  to  the 
measurement  of  resistances  of  the  same  order  of  magnitude  but 
smaller  than  the  total  resistance  of  the  rheostat.  When  the 
instrument  maker  can  be  trusted  to  accurately  adjust  the  ratio 


ART.  509] 


WHEATSTONE-BRIDGE   METHODS 


91 


coils  in  a  Wheatstone  bridge  the  reversible  feature  is  scarcely 
worth  its  extra  cost  as  applied  to  the  "  Post  Office  "  type  of 
bridge. 

509.  Schone's  Arrangement  of  Ratio  Arms.  —  This  very 
superior  disposition  of  ratio  coils  was  described  by  0.  Schone, 
in  "  Zeitschrift  fur  Instrumentenkunde,"  May, 
1898.  It  is  now  extensively  used  in  America  and 
by  its  superiority  deserves  to  supersede  all  other 
arrangements  of  resistances  for  reversible  ratio  arms. 

According  to  this  arrangement  all  the  ratio  coils 
have  one  of  their  terminals  joined  to  a  common 
bar  connector  which  corresponds  to  the  block 
marked  C  of  Fig.  508a.  The  other  terminal  of 
each  coil  is  joined  to  a  separate  block.  The  scheme 
is  given  in  Fig.  509. 

The  bar  A  on  one  side  of  these  blocks  is  joined 
to  the  rheostat  R,  and  the  bar  B,  on  the  other  side, 
to  an  X  post. 

In  the  ordinary  use  of  this  arrangement  two  plugs 
only  are  used.  One  plug  is  inserted  between  the 
bar  A  and  one  of  the  blocks  1,  1',  10,  10',  etc.,  of 
the  central  row,  and  the  other  plug  is  inserted  be- 
tween the  bar  B  and  any  one  of  the  blocks  of  the 
central  row,  except  the  one  which  the  other  plug 
joins  to  bar  A. 


100 


100 


1000 


10000 


FIG.  509. 


The  construction  generally  embodies  two  ratio  coils  of  each 
value.  Referring  to  Fig.  509,  if  one  wishes  to  obtain  a  unity 
ratio,  as  1000  to  1000',  one  plug  would  be  inserted  between  the 
block  1000  and  the  bar  A ,  and  the  other  plug  between  the  block 
1000'  and  the  bar  B.  This  disposition  of  the  plugs  joins  the  end 
of  the  1000  ohm  coil  to  the  rheostat  and  the  end  of  the  1000'  ohm 
coil  to  the  X  post.  If,  now,  one  plug  is  inserted  between  the 
1000'  block  and  bar  A,  and  another  plug  between  the  1000  block 
and  bar  B,  the  ratio  arms  become  reversed;  that  is,  the  1000'  ohm 
coil  is  joined  to  the  rheostat,  and  the  1000  ohm  coil  to  the  X 
post. 

When  uneven  ratios  are  used  the  same  ratio  can  be  obtained  by 
four  different  combinations.  If  we  wish  to  obtain  the  ratio  1  to 
10,  we  can  plug  between  A  and  1  and  B  and  10  and  get  1  to  10, 
or  between  A  and  1'  and  B  and  10  and  get  1'  to  10,  or  between 


92 


MEASURING  ELECTRICAL  RESISTANCE        [Aux.  510 


A  and  1  and  B  and  10'  and  get  1  to  10',  or  between  A  and  1' 
and  B  and  10'  and  get  I'  to  10'. 

To  obtain  the  reciprocal  set  of  ratios,  like  the  above,  we  would 
plug  A  and  10,  B  and  1,  and  get  10  to  1;  A  and  10',  B  and  1, 
and  get  10'  to  1;  A  and  10,  B  and  1',  and  get  10  to  1';  A  and  10', 
B  and  1',  and  get  10'  to  1'. 

By  using  more  than  two  plugs  and  connecting  certain  of  the 
coils  in  parallel  combinations,  a  large  number  of  other  ratios  may 
be  obtained.  For  example,  we  can  plug  between  A  and  100  and 
A  and  100',  and  between  B  and  1000  and  get  the  ratio  50  to  1000, 
or  we  can  plug  between  A  and  1000  and  A  and  1000'  and  between 
B  and  100  and  get  the  ratio  500  to  100. 

With  this  arrangement  of  ratio  coils  it  is  seen  that  errors  due 
to  plug  contacts  become  practically  nil,  because  only  two  plug 
contacts  enter  the  circuit,  while  with  even  ratios  it  is  only  the 
difference  in  the  resistance  of  the  two  plug  contacts  which  affects 
the  results. 

510.  Nonreversible  Ratio  Arms  Adjustable  without  Contact 
Resistances.  —  A  very  excellent  arrangement  of  variable,  but 


FIG.  510. 

nonreversible  ratio  arms,  which  involves  no  contact  resistances 
in  the  ratio  arms,  is  shown  in  Fig.  510.  The  ratio  values  may  be 
varied  by  moving  a  brush  contact,  which  is  joined  to  the  battery, 
over  studs  as  indicated  in  the  figure. 

To  calculate  the  odd  values  required  for  ratio  coils  the  solution 
must  be  found  for  equations  of  the  form, 


ART.  511]  WHEATSTONE-BRIDGE    METHODS  93 

a  1 


b+c+d+e+f   100 
a  +  b      _!_ 
~  10' 


a  +  b  +  c 
d+e+f= 
b  +  c  +  d  _1A 


e+f 

+e 

~  —  1UU. 


In  the  case  selected  we  have  five  equations  and  six  unknowns, 
hence  some  one  of  the  quantities,  as  a,  must  be  assumed  as  known. 
If  we  choose  a  =  1  ohm,  then  the  solution  of  the  above  equations 
gives 

a  =  l,    6  =  8.1818,    c  =  41.3182,    d  =  41.3182,    e  =  8.1818,    /  =  !. 

We  note  that  a  =  /,  b  =  e,  and  c  =  d,  and  therefore  there  are  but 
two  odd  values  of  resistance  to  adjust  to  give  the  five  different 
ratio  settings,  100,  10,  1,  0.1,  0.01. 

The  method  may  be  indefinitely  extended  and  is  an  excellent 
arrangement  to  use  with  bridges  in  which  the  rheostat  values  are 
varied  by  means  of  dials  and  sliding-brush  contacts,  for  then  all 
resistance  changes  in  the  box  can  be  effected  with  dials  and  no 
plugs  are  required.  By  this  arrangement  the  ratio  arms,  being 
free  from  contact  resistances,  give  ratios  just  as  accurately  as  the 
coils  are  adjusted. 

511.  Wheatstone  Bridge  Arranged  for  Reading  in  Per  Cent.  — 
It  frequently  happens  that  the  problem  of  very  rapidly  measuring 
a  large  number  of  resistance  units  of  even  values  as  1,  10,  100 
ohms,  etc.,  is  presented.  Instrument  makers  have  this  to  do  in 
checking  up  the  precision  of  the  coils  in  resistance  boxes.  In  such 
cases  it  is  of  little  interest  to  know  the  absolute  number  of  ohms 
by  which  any  coil  is  in  error,  the  important  question  being  what  is 
the  per  cent  accuracy  of  any  coil. 

To  meet  the  above  requirements  of  a  Wheatstone  bridge,  the 
author  devised  the  method  and  connections  given  below  in  Fig.  511. 
Bridges  were  constructed,  embodying  the  connections  of  Fig.  511 
and  placed  in  continuous  service,  which  would  give  by  a  direct 
reading  the  per  cent  value  of  any  coil  being  measured  in  terms 
of  the  standard  employed,  The  readings  of  the  last  of  the  four 


94 


MEASURING   ELECTRICAL   RESISTANCE          [ART.  512 


dials  used  were  in  steps  of  0.001  of  1  per  cent,  and  the  range  of 
the  bridge  was  from  95  per  cent  to  106  per  cent  of  the  standard. 

As  a  method  may  be  applied  for  eliminating  the  lead  resistances, 
the  per  cent  bridge  may  be  used  advantageously  for  resistances  from 
1  ohm  up,  with  errors  not  exceeding  0.001  of  1  per  cent. 

The  diagram,  Fig.  511,  is  practically  self-explanatory. 


1-   500^- Spool 


FIG.  511. 


To  use  this  method,  one  first  connects  the  lead  wires  together, 
which  go  to  the  coil  to  be  measured.  All  the  bridge  dials  are  set 
so  as  to  read  an  even  100  per  cent.  Then  the  lead  wires  which 
go  to  the  standard  resistance  are  also  joined  together  at  the  ends 
which  connect  to  the  standard,  and  they  are  varied  in  length  until 
the  galvanometer  shows  a  balance.  This  means  that  the  lead 
resistances  to  the  X  coil  are  equal  to  the  lead  resistances  to  the 
standard  coil,  and  as  the  X  coil  is  never  very  greatly  different  in 
value  from  the  standard,  the  lead  resistances  eliminate. 

The  bridge  requires  for  its  practical  use  a  complete  set  of  re- 
sistance standards,  against  which  to  match  the  resistance  coils  to 
be  measured. 

This  form  of  the  Wheatstone  bridge  is  of  great  value  to  the 
instrument  maker  who  has  many  coils  to  measure  with  both 
rapidity  and  precision. 

512.  Remarks  upon  the  Use  of  the  Wheatstone  Bridge.  —  In 
arranging  to  use  a  Wheatstone  bridge  with  accuracy,  speed,  and 
convenience,  one  should  select  with  care  a  suitable  galvanometer 
or  other  detector  for  indicating  when  the  bridge  is  balanced.  In 
addition  to  its'  greater  sensibility  a  galvanometer  is  more  ad- 
vantageous than  detectors  of  the  telephone  type,  in  that  the  deflec- 


ART.  512]  WHEATST  ONE-BRIDGE   METHODS  95 

tions  are  to  the  right  or  to  the  left,  according  as  the  rheostat  adjust- 
ment is  higher  or  lower  than  the  setting  required  for  a  balance; 
whereas  with  the  telephone  the  sound  increases  equally  and  with- 
out distinction  for  a  departure  from  the  setting  of  the  rheostat  in 
either  direction  from  that  which  gives  a  true  balance.  Further- 
more, when  a  telephone  is  used,  the  current  through  the  bridge  arms 
must  be  made  variable  to  cause  a  sound  in  the  telephone,  and 
correct  values  of  resistance  will  only  be  obtained  by  meeting  the 
condition,  not  always  possible  of  fulfillment,  that  the  four  arms 
of  the  bridge  are  without  appreciable  capacity  or  inductance. 
These  considerations  practically  necessitate,  for  the  general  use 
of  the  Wheatstone  bridge,  the  employment  of  a  galvanometer  as 
the  instrument  to  show  when  the  bridge  is  balanced. 

The  variety  and  the  types  of  bridges  and  the  methods  of  their 
employment  are  so  great  that  no  general  rules  can  be  laid  down 
as  to  what  kind  of  galvanometer  will  best  serve  the  purpose.  How- 
ever, a  few  general  considerations  may  be  mentioned.  Except  for 
special  requirements,  a  mirror  galvanometer  of  the  D'Arsonval 
type,  having  a  resistance  of  from  100  to  500  ohms,  will  be  found 
convenient,  and  will  have  ample  sensibility  if  it  will  show  a  deflec- 
tion of  one  division  on  a  scale  1000  divisions  from  the  mirror  with  a 
current  of  0.005  microampere.  The  galvanometer  should  be  just 
aperiodic  to  save  time  in  waiting  for  the  deflections  to  return  to 
zero  and  increased  satisfaction  will  be  found  in  working  in  pro- 
portion as  the  period  of  the  galvanometer  is  made  shorter.  A 
galvanometer  of  the  D'Arsonval  type  having  a  period  of  three 
seconds,  a  resistance  of  200  ohms  and  a  sensibility  of  200  megohms, 
can  easily  be  constructed  and  will  admirably  meet  nearly  all  the 
requirements  of  Wheatstone-bridge  work  of  high  precision.  It 
should  be  recalled  that  a  galvanometer  sensibility  is  expressed  in 
megohms  when,  with  the  scale  at  1000  scale  divisions  from  the 
mirror,  the  sensibility  Sm  is  the  number  of  megohms  which  must 
be  in  the  galvanometer  circuit,  so  that,  with  an  E.M.F.  of  1  volt 
in  the  circuit;  there  will  result  a  deflection  of  one  scale  division. 
One  should  distinguish  sensibility  from  "  figure  of  merit  "  which 
is  defined  by  the  equation 


*  See  par.  1504,  Eq.  (11),  also  article  by  Edwin  F.  Northrup  in  the  Journal 
of  the  Franklin  Institute,  Oct.,  1910,  entitled  "The  Comparison  of  Galvanom- 
eters and  a  New  Type  of  Flat-coil  Galvanometer." 


96  MEASURING  ELECTRICAL  RESISTANCE        [ART.  512 

where  T  is  the  undamped  complete  period  of  the  galvanometer 
and  R  the  resistance  of  its  coil. 

For  most  uses  of  the  slide-wire  Wheatstone  bridge  and  other 
types  in  which  the  coils  are  adjusted  to  an  accuracy  of  not  better 
than  0.05  of  1  per  cent,  it  is  unnecessary  to  use  a  reflecting  type 
of  galvanometer,  with  either  telescope  and  scale  or  lamp  and 
scale.  A  small  pointer  galvanometer,  of  100  or  200  ohms 
resistance,  having  a  sensibility  such  that  with  1  volt  and 
250,000  ohms  in  circuit  the  pointer  will  deflect  1  millimeter  on 
its  scale,  will  be  found  amply  sensitive  and  very  convenient  to 
use. 

The  relative  positions  of  the  battery  and  the  galvanometer  in 
the  Wheatstone-bridge  circuits  should  be  chosen  to  meet  the 
condition  that  the  terminals  of  the  galvanometer  shall  connect 
such  junction  points  of  the  four  arms  of  the  bridge  as  will  make 
as  nearly  as  possible  the  resistance  external  to  the  galvanometer 
equal  to  the  resistance  of  the  galvanometer  itself.  For  example, 
if  the  resistance  X  is  10  ohms,  and  the  ratio  arms  are  made  100 
ohms  to  1  ohm,  the  terminals  of  the  galvanometer,  if  this  has  a 
resistance  of  100  ohms  or  more,  should  be  connected,  one  to  the 
junction  point  of  the  100  with  the  1000  ohms  coil  of  the  rheostat, 
and  the  other  to  the  junction  point  of  the  1  with  the  10  ohms  coil. 
Maxwell  gives,  in  his  "  Electricity  and  Magnetism,"  Vol.  I,  par. 
348,  the  following  rule:  "Of  the  two  resistances,  that  of  the  bat- 
tery and  that  of  the  galvanometer,  connect  the  greater  resistance  so 
as  to  join  the  two  greatest  to  the  two  least  of  the  four  resistances." 
In  modern  practice,  one  generally  uses  a  battery  of  4  or  6  volts,  and 
then  reduces  the  current  in  the  bridge  circuit  to  a  suitable  value 
by  the  use  of  a  resistance  in  series  with  the  battery,  and  the  posi- 
tion occupied  by  the  galvanometer  should  be  chosen  with  refer- 
ence to  its  own  resistance  only,  as  compared  with  the  resistances 
of  the  bridge  arms.  The  object  to  be  obtained  is  that  the  circuit 
external  to  the  galvanometer  should  be  as  nearly  as  possible  that 
of  the  galvanometer  itself,  without  regard  to  the  battery  resistance. 
Hence  the  rule  at  the  beginning  of  the  paragraph.  The  object  for 
this  choice  of  position  of  the  galvanometer  is  to  give  the  arrange- 
ment the  maximum  sensibility,  but,  with  the  current-carrying 
capacity  of  the  manganin  coils  now  in  use,  of  practically  zero 
temperature  coefficient,  and  with  the  high  sensibility  of  easily 
obtainable  galvanometers,  the  sensibility  is  generally  adequate 


ART.  512]  WHEATSTONE-BRIDGE   METHODS  97 

however  the  position  of  the  galvanometer  is  chosen,  and  the  im- 
portance of  fulfilling  the  above  conditions  is  slight. 

The  safe  watt  capacity  of  the  coils  of  a  Wheatstone  bridge  will 
vary  from  one-quarter  to  four  watts  per  coil  according  to  its  con- 
struction. If  this  watt  capacity  of  a  coil  is  greatly  exceeded  the 
coil  may  be  heated  to  a  point  where  the  resistance  is  permanently 
changed,  even  though  the  insulation  is  not  charred.  It  should 
always  be  remembered  that  the  watt  load  put  on  any  coil  in  a 
bridge  is  equal  to  the  square  of  the  potential  applied  at  its  ter- 
minals divided  by  its  resistance,  and  that,  as  a  rule,  this  quantity 
should  never  exceed  1  watt.  Unless  one  should  forget  and  make 
connections  which  would  bring  an  excessive  voltage  at  the  ter- 
minals of  a  coil,  it  is  always  well  in  order  to  avoid  this  danger  to 
keep  an  external  resistance  in  circuit  with  the  battery.  This  will 
limit,  for  any  connections  of  the  bridge,  the  flow  of  current  to  a 
safe  amount. 

Even  though  the  rheostat  of  a  bridge  is  incapable  of  being  varied 
by  very  small  steps,  one  can  measure  resistances  with  exactness 
by  making  use  of  the  deflections  of  the  galvanometer  after  a 
balance  has  been  obtained  within  an  adjustment  of  the  smallest 
step  of  the  rheostat.  The  procedure  is  as  follows:  The  current 
furnished  by  the  battery  being  assumed  constant  and  the  deflec- 
tions of  the  galvanometer  proportional  to  the  current  through  it, 
one  takes  note  of  the  permanent  deflection  of  the  galvanometer 
when  the  resistance  of  the  rheostat  required  for  a  balance  is  set 
too  small,  in  a  final  adjustment,  by  the  smallest  step  in  the  rheo- 
stat. Call  R  this  resistance  and  d  the  deflection.  Then  increase 
the  resistance  of  the  rheostat  by  one  of  its  smallest  steps,  say  one 
ohm,  and  observe  the  deflection  then  obtained  which  will  be  in 
the  opposite  direction  to  the  one  previously  obtained.  Call  this 
deflection  df.  The  true  value  of  X  will  then  be  given  by  the 
relation 


where  a  and  b  are  the  resistances  in  the  ratio  arms,  and  s  the 
value  in  ohms  of  the  smallest  step  in  the  rheostat. 

When  the  resistance  to  be  measured  is  wholly  unknown  one 
should  proceed,  in  seeking  a  balance,  in  a  systematic  manner.  To 
avoid  violent  deflections  of  the  galvanometer  this  may  be  tempo- 
rarily shunted  with  a  low  resistance,  which  shunt  is  removed  when 


98  MEASURING  ELECTRICAL  RESISTANCE         [ART.  512 

a  balance  is  nearly  obtained.  It  is  well  to  start  with  unity  ratio 
and  with  zero  resistance  in  the  rheostat.  A  quick  tap  of  the  key 
will  cause  a  moderate  deflection  of  the  shunted  galvanometer  in 
one  direction.  1000  ohms  may  now  be  put  in  the  rheostat  and 
the  key  be  again  tapped.  A  deflection  in  the  opposite  direction 
will  now  indicate  that  the  resistance  lies  between  0  and  1000  ohms. 
500  ohms  should  now  be  plugged  in  the  rheostat  and,  if  the  deflec- 
tion is  like  the  first  one  when  the  key  is  tapped,  the  resistance  is 
known  to  lie  between  500  and  1000  ohms. 

By  proceeding  in  this  manner  the  resistance  is  narrowed  down, 
with  only  a  few  trials,  very  close  to  its  actual  value.  One  should 
now  choose  the  value  of  the  ratio  so  that  in  obtaining  a  final 
balance  the  largest  possible  portion  of  the  rheostat  is  brought  into 
service.  The  final  balancing  is  made  with  the  shunt  removed 
from  the  galvanometer  and  the  procedure  to  be  followed  is  pre- 
cisely that  adopted  for  weighing  with  a  delicate  balance.  With 
a  galvanometer  in  which  the  coil  is  visible  the  preliminary  balanc- 
ing is  usually  effected  by  directly  observing  the  movements  of  the 
coil.  In  the  final  adjustments  only  is  it  necessary  to  observe  the 
movements  of  the  coil  by  looking  thru  the  telescope,  or  by  ob- 
serving the  spot  of  light  on  the  scale. 

No  definite  limitations  can  be  laid  down  for  the  useful  resist- 
ance range  of  a  Wheatstone  bridge,  as  this  depends  upon  the 
range  of  its  rheostat  and  upon  the  number  and  precision  of  the 
coil-values  provided  in  its  ratio  arms.  Ordinarily,  Wheatstone 
bridges  should  be  considered  adaptable  for  the  fairly  accurate 
measurement  of  resistances  which  lie  between  1  and  1,000,000 
ohms,  though  this  range  is  often  exceeded  in  both  directions  with 
high-class  bridges. 

The  precision  of  measurements  possible  with  a  Wheatstone 
bridge  depends  upon  a  variety  of  circumstances,  such  as,  the 
value  of  the  resistance  being  measured,  the  accuracy  of  the  coils 
in  the  bridge,  the  possibility  of  reversing  the  ratio  arms  to  elimi- 
nate their  error,  and  the  care  with  which  contact  resistances  are 
allowed  for,  or  guarded  against.  In  routine  work,  for  resistances 
of  the  same  order  of  magnitude  as  the  total  resistance  of  the 
bridge  rheostat,  a  precision  of  0.04  of  1  per  cent  may  be  considered 
fairly  good,  tho  the  author  owns  a  bridge  which  can  be  relied 
upon  to  measure  resistances  in  the  range  from  10  to  10,000  ohms 
to  an  accuracy  better  than  0.02  of  1  per  cent. 


ART.  512]  WHEATSTONE-BRIDGE  METHODS  99 

Since  the  advent  of  manganin  coils  with  their  practically  zero 
temperature  coefficients,  little  regard  need  be  given  to  the  interior 
temperature  of  the  bridge.  The  temperature  of  the  resistance 
being  measured,  however,  unless  this  is  also  of  a  zero  temper- 
ature coefficient  material,  must  be  carefully  observed. 


CHAPTER  VI. 
THE  MEASUREMENT  OF  LOW  RESISTANCE. 

600.  Introductory  Statement.  —  When  one  is  about  to  make 
an  electrical  measurement,  it  is  often  not  possible  to  choose  the 
best  method  because  the  apparatus  for  this  is  not  available. 
For  this  reason  it  is  desirable  to  be  acquainted  with  alternative 
methods,  and  this  consideration  leads  us  to  describe  several 
methods  for  measuring  low  resistances,  tho  the  one  known  as  the 
"  Kelvin-double-bridge  "  method  is  preeminently  the  most  accu- 
rate and  elegant,  and,  when  apparatus  suitable  for  its  application 
is  to  be  had,  should  be  chosen  in  preference  to  any  other. 

While  there  is  no  sharp  distinction  between  medium  and  low 
resistance  we  may,  for  convenience,  consider  any  resistance  which 
is  less  than  one  ohm  as  low.  Ordinary  methods,  applicable  to 
medium  resistances,  fail  to  give  precision  with  low  resistances 
either  on  account  of  contact  resistances  which  are  likely  to  enter 
the  circuit  which  contains  the  resistance  being  measured  or  be- 
cause a  low  resistance  is  often  a  short  conductor,  and  errors  in  the 
determination  of  the  exact  length  measured  are  apt  to  enter. 
Both  these  sources  of  error  are  avoided  by  providing  the  re- 
sistance measured  and  the  standard  with  which  it  is  compared 
with  potential  points.  The  resistance  which  is  determined  is  the 
resistance  which  lies  between  two  potential  points,  when  the  lines 
of  current  flow  thru  the  low  resistance  have  a  particular  distribu- 
tion. It  should  be  remarked,  that,  if  the  resistances  of  several 
conductors  are  given,  each  provided  with  fixed  potential  points 
to  which  connections  may  be  made,  these  conductors  cannot  be 
joined  in  series  or  in  parallel  combinations  to  obtain  a  known 
resultant  resistance.  For  this  reason  standards  of  resistance, 
provided  with  fixed  potential  points,  are  unsuited  for  obtaining 
other  values  by  series  or  parallel  combinations. 

For  the  measurement  then  of  low  resistance  of  widely  varying 
range,  one  needs  to  be  provided  with  a  series  of  low-resistance 

100 


ART.  601]      THE   MEASUREMENT  OF  LOW  RESISTANCE  101 

precision  standards.  A  set  of  precision  standards  which  would 
be  quite  complete  would  consist  of  1,  0.1,  0.01,  0.001,  0.0001 
ohm,  the  resistance  values  being  in  every  case  between  fixed 
potential  points.  The  standards  would  each  have,  therefore, 
two-current  and  two-potential  terminals.  They  are  usually 
mounted  in  metal  cylindrical  boxes  which  can  be  filled  with  kero- 
sene or  paraffin  oil  to  permit  of  an  accurate  determination  of  their 
temperatures.  They  are  made  usually  of  manganin  wire  or  sheet, 
which  alloy,  consisting  of  nickel,  copper  and  manganese,  has  a 
low  temperature  coefficient  which  may  be  taken  roughly  as 
0.00002  per  ohm  per  degree  centigrade.  Its  thermoelectric  force 
against  copper  is  also  very  small,  which  is  of  importance  for  many 
classes  of  measurements. 

The  catalogues  of  Otto  Wolff  of  Berlin  and  of  The  Leeds  and 
Northrup  Company  of  Philadelphia,  Pa.,  give  very  full  informa- 
tion upon  the  best  types  of  low-resistance  standards  upon  the 
market,  as  well  as  upon  the  apparatus  especially  adapted  to  the 
measurement  of  low  resistance  by  null  methods. 

601.  Low  Resistance  Measured  with  an  Ammeter  and  a 
Millivoltmeter.  —  Join  the  conductor  ab,  the  resistance  of  which 

9 

/TNM.V. 


^i^ 

7 -J 

i          —n 


R  P, 

A 


is  to  be  determined  between  two  points,  in  series  with  a  rheostat  r , 
an  ammeter  A  and  one  or  two  cells  of  storage  battery  Ba  (Fig.  601) . 
To  the  two  points  p  and  pi  between  which  the  resistance  is  to  be 
determined,  join  the  terminals  of  a  millivoltmeter  by  means  of 
knife-edges  pressed  upon  the  conductor.  The  length  I  between 
these  knife-edge  contacts,  which  become  potential  points,  should 
be  accurately  measured.  The  rheostat  r  should  now  be  adjusted 
until  both  the  ammeter  and  the  millivoltmeter  give  suitably  high 
readings  on  their  scales.  Then  calling  V  the  reading  of  the  milli- 


102  MEASURING  ELECTRICAL  RESISTANCE         [ART.  602 

voltmeter,  /  the  reading  of  the  ammeter,  g  the  resistance  of  the 
millivoltmeter,  and  R  the  resistance  to  be  measured,  we  have 
gR         7  V 

I 

g 

v 

If  the  resistance  R  is  a  very  low  resistance  the  quantity  —  may 

y 

be  neglected  without  appreciable  error.  An  example  may  be 
found  in  the  determination  of  the  resistance  of  one  meter  of  No.  0 
aluminum  wire.  The  resistance  of  this  at  20°  C.  would  be  about 
0.00052  ohm.  Hence  20  amperes  would  give  a  drop  of  0.0104 
volt  or  10.4  millivolts  over  the  wire  itself.  We  would  have  by  the 

0  0104 
exact  formula,  if  g  =  1  ohm,  R  =  *"         =  0.00052027  ohm. 

20-    ^ 

By  neglecting  the  last  term  of  the  denominator  an  error  of  about 
0.05  of  1  per  cent  only  would  be  committed.  In  most  measure- 
ments of  this  kind  the  chief  source  of  error  will  result  from  a 
neglect  to  accurately  determine  the  temperature  of  the  sample, 
the  resistance  of  which,  if  a  pure  metal,  will  vary  about  0.4  of 
1  per  cent  per  degree  centigrade.  It  should  be  remarked  that  in 
this  method  the  resistance  is  not  compared  with  another  resist- 
ance taken  as  a  standard,  but  is  measured  by  two  instruments 
upon  the  accuracy  of  the  calibration  of  which  the  accuracy  of 
the  measurement  depends. 

If  a  standard  of  low  resistance,  of  approximately  the  same 
magnitude  as  the  resistance  to  be  measured,  is  available,  then  the 
more  precise  method  would  be  to  join  the  two  resistances  in  series 
and  (still  using  the  ammeter  to  test  the  constancy  of  the  current) 
read  with  the  millivoltmeter  the  drop,  first  over  the  standard  and 
then  over  the  sample.  The  resistances  are  then  in  the  same  ratio 
as  the  drops  of  potential,  provided  the  current  taken  by  the  milli- 
voltmeter is  either  negligibly  small  compared  with  the  main 
current  or  that  the  standard  resistance  is  closely  the  same  as  the 
resistance  being  measured. 

602.  To  Measure  the  Resistance  of  Sections  of  a  Closed  Cir- 
cuit; General  Method.  —  The  problem  is  often  presented  in 
commercial  practice  of  obtaining  the  resistance  of  a  portion  of  a 
circuit  which  is  closed  upon  itself  and  which  may  contain  a  source 
of  current,  either  alternating  or  direct.  If  the  circuit  could  be 


ART.  602]      THE   MEASUREMENT   OF  LOW  RESISTANCE 


103 


opened  even  momentarily  the  problem  could  be  solved  by  well- 
known  methods.  But  if  the  circuit  cannot  be  opened,  the  problem 
is  still  solvable  in  more  than  one  way.  The  following  methods  have 
been  independently  devised*  by  the  writer  and  carefully  tested 
out  in  practical  cases,  and  have  been  found  to  give  such  satis- 
factory results  as  to  warrant  a  detailed  description. 

We  shall  first  consider  a  general  method  applicable  to  measur- 
ing the  resistance  of  a  closed  conductor  or  loop,  such  as  the  rim 
of  a  cart-wheel,  which  may  be  assumed  to  have  a  cross-section 
which  varies  in  an  unknown  way  from  one  portion  of  its  circum- 
ference to  another.  Referring  to  Fig.  602,  we  have  the  follow- 
ing dispositions  of  circuits  and  instruments. 


M.V.  No.  3 


M.V.  No.  2 
Reads  E2'orE2 


FlG.  602. 

P  is  a  closed  metallic  circuit  of  medium  or  very  low  resistance 
which  cannot  be  opened.  It  is  required  to  determine  the  resistance 
#  of  a  definite  length  of  this  closed  circuit,  as  between  two  points 
a  and  b.  For  this  there  are  required  three  deflection  instruments 
which  deflect  proportionally  to  the  current  thru  them.  The 
constant  of  these  instruments  need  not  be  known  but,  if  not  known, 
the  value  must  be  the  same  for  all  three.  In  the  present  applica- 
tion of  the  method  there  is  required  one  known  resistance  R 
provided  with  potential  terminals.  This  resistance  R  should  be 
chosen,  for  the  best  accuracy,  of  the  same  order  of  magnitude  as 
the  resistance  x  which  is  to  be  determined.  A  cell  of  storage 

*  The  original  development  of  methods  of  measuring  the  resistance  of  closed 
circuits  and  the  current  in  underground  mains  is  due  to  Dr.  Carl  Hering.  See 
his  paper  and  author's  discussion  presented  at  the  20th  Annual  Convention 
of  tfie  American  Inst.  of  Elec.  Engs.,  Boston,  Mass.,  June  28,  1912. 


104  MEASURING  ELECTRICAL  RESISTANCE         [ART.  602 

battery  and  a  rheostat  r  to  adjust  the  current  from  the  battery  to 
a  suitable  value  are  required,  also  a  key  K.  The  deflection  instru- 
ments would  ordinarily  be  millivoltmeters,  tho  three  galvanom- 
eters having  the  same  constant  could  be  used.  The  M.V.  No. 
1  is  joined  to  the  potential  terminals  a,  6,  between  which  the 
resistance  is  x.  The  M.  V.  No.  2  is  joined  to  the  potential  ter- 
minals/, d,  between  which  the  resistance  is  y,  and  the  terminals 
of  the  M.V.  No.  3  are  joined  to  the  potential  terminals  q,  s,  be- 
tween which  the  resistance  is  R,  which  is  known.  The  current 
terminals  of  R  are  joined  to  the  points  p,  g  of  the  loop  and  in 
circuit  with  R  is  the  key  K.  The  cell  of  storage  battery  Ba,  which 
includes  in  its  circuit  the  rheostat  r,  is  joined  to  two  points,  as  m 
and  n,  of  the  closed  metallic  circuit.  This  supplies  to  the  system 
the  current  required  for  the  measurement. 

The  procedure  in  making  a  measurement  is  as  follows  : 

(a)  With  the  key  K  open,  read  at  the  same  moment  the  V.M. 
No.  1  and  call  its  deflection  di   and  the  V.M.  No.  2  and  call  its 
deflection  d%  '. 

(b)  With  the  key  K  closed  read  simultaneously  the  three  deflec- 
tion instruments.     Call  the  deflection  of  V.M.  No.  1,  di,  of  V.M. 
No.  2,  d2  and  of  V.M.  No.  3,  D.     Then  in  case  (a) 

-  =  %  ,  which  call  N,     Then 
y     d2 

x  =  Ny.  (1) 

In  case  (b),  since  the  deflections  D,  di,  and  d2  are  proportional 
respectively  to  E.M.F.'s  7,  E1}  and  Ez,  we  have 

and                                 E^Kd^dx  (2) 

E2==Kd2==Cy.  (3) 

Here  K  is  a  constant  and  C  is  the  current  thru  y,  and  C\  is  'the 
current  thru  x.  We  also  have 

C  =  Ci  +  /, 
where  /  is  the  current  thru  R.    But 

7      KD 

R  =  ~lt~  '    whence> 


In  the  relations  (1),  (2),  (3),  (4),  we  have  the  unknown  quantities, 


ART.  602]      THE   MEASUREMENT   OF  LOW  RESISTANCE  105 

x,  y,  C,  Ci,  and  hence,  there  being  but  four  unknowns  and  four 
equations,  both  x  and  y  can  be  determined. 
We  finally  derive 

dzN  -dlD  _ 

x  =      —  p  -  R,  (5) 

and  y  =  %x.  (6) 

ai 

Equation  (5)  is  obtained  as  follows: 
From  Eqs.  (3)  and  (4) 


From  Eqs.  (2)  and  (7) 

Kdl      Kd2  _  KD 
x  y          R' 

di      d2      D 
¥=y-r 

Putting  in  Eq.  (8)  the  value  of  y  from  Eq.  (1),  we  obtain 

dl_d*N_D 

¥"  T"     R9 

and  from  Eq.  (9)  we  find  the  value  of  x  to  be  that  given  in  Eq.  (5). 

The  above  method  possesses  four  special  merits:  The  circuit 
of  the  resistance  being  measured  does  not  have  to  be  opened; 
the  resistance  of  no  contact  enters  and  hence  the  contacts  at 
points  p,  g,  m,  n,  and  K  need  not  be  made  with  any  special  care, 
while  the  points  a,  6,  /,  d,  q,  and  s  are  merely  potential  points 
and  contact  at  these  places  may  be  made  with  a  sharp  point  or 
knife-edge  pressed  against  the  conductor;  the  constant  of  the 
deflection  instruments  need  not  be  known,  it  being  only  necessary 
that  all  three  instruments  have  the  same  constant;  two  instru- 
ments are  read  in  case  (a)  simultaneously  and  three  instruments 
are  read  simultaneously  in  case  (b),  and  hence  the  current  in  the 
loop  P  may  be  very  variable  and  accurate  results  still  be  obtained. 

This  method  was  tried  by  the  author,  using  a  brass  ring  a  little 
over  one  meter  in  circumference  and  of  No.  0  B.  &  S.  wire.  The 
ring  was  placed  over  an  open-core  alternating-current  electromag- 
net of  very  great  size.  By  exciting  the  alternating-current  magnet 
induced  alternating  currents  were  sent  thru  the  ring.  It  was 
found  that  the  readings  of  the  three  instruments,  and  hence  the 


106  MEASURING   ELECTRICAL  RESISTANCE         [ART.  603 

resistance  measured,  was  in  no  wise  affected  by  the  presence  of 
the  alternating  current  induced  in  the  ring,  hence  the  method 
applies  whether  the  closed  loop  is  or  is  not  carrying  an  alternating 
current. 

In  the  above  trial  the  actual  readings  observed  and  the  results 
obtained  were  as  follows: 

di  =  20.54,     d2  =  25.18,     D  =  18.51,     R  =  0.01  ohm. 
The  ratio  of  d\    to  d2'j  or  N,  was  0.9940.     From  these  readings 
the  value  obtained  for  x  was,  by  Eq.  (5), 

25.18  X  0.994  -  20.54  . 
x  =  -    T^— X  0.01  =  0.002425  ohm. 

lo.Ol 

The  ring  was  afterward  cut  open  and  the  resistance  x  was  deter- 
mined by  an  ordinary  method  and  found  to  be  0.002439  ohm. 
Hence,  the  error  in  the  measurement  of  the  closed  ring  was  0.57 
of  1  per  cent. 

This  method  has  a  useful  application  when  applied  to  the 
determination  of  the  resistance  between  two  points  in  a  bus- 
bar while  this  is  carrying  direct  current. 

603.  To  Measure  the  Resistance  Between  Two  Points  on  a 
Bus-bar.  —  The  arrangement  to  employ  is  represented  in  Fig.  603. 


r 


FIG.  603. 

The  potential  points  a,  b  and  c,  d  may  be  obtained  by  drilling 
and  tapping  small  holes  in  the  bus-bar  and  inserting  in  these  holes 
small  screws  to  which  the  terminals  of  the  millivoltmeters  may 
be  secured.  The  terminals  of  the  resistance  R  may  be  attached 
to  the  bus-bar  at  p  and  q  by  means  of  iron  clamps,  as  the  precision 
of  the  method  is  not  affected  by  contact  resistances  at  these  places. 
The  distances  between  the  point  a  and  the  clamp  p  and  the  point 
b  and  the  clamp  q  should  be  at  least  three  times  the  width  of  the 


ART.  603]      THE   MEASUREMENT  OF  LOW  RESISTANCE  107 

bus-bar.  It  is  also  desirable  to  have  the  clamps  p  and  q  make 
contact  with  the  bus-bar  across  its  entire  width.  The  purpose 
of  these  two  precautions  is  to  insure  that  the  stream  lines  of  current 
are  parallel  with  the  bus-bar  at  the  potential  points  a  and  b. 
For  the  same  reason  the  potential  point  c  should  be  as  far  to  the 
right  of  q  as  the  potential  point  b  is  to  the  left.  The  distance  from 
c  to  d  should  be  chosen  about  equal  to  the  distance  from  a  to  b  in 
order  to  bring  the  ratio  N  near  unity. 

If  there  is  direct  current  in  the  bus-bar,  supplied  by  the  genera- 
tor, then  there  is  no  necessity  of  introducing  additional  current 
from  storage  cells,  as  is  required  when  measuring  the  resistance 
of  a  section  of  a  loop,  as  described  in  par.  602. 

The  standard  resistance  R  should  be  supplied  with  potential 
points  and  should  be  not  over  ten  times  the  resistance  of  the  bus- 
bar between  the  clamps  p  and  q.  Greater  accuracy  will  be  ob- 
tained if  this  resistance  is  about  equal  to  the  resistance  of  the 
bus-bar  between  the  clamps.  Since  the  drop  of  potential  over 
the  resistance  R  is  read  to  give  the  value  of  the  current  /  which 
flows  in  the  branch  circuit,  one  may  substitute  an  ammeter  for 
the  resistance  R  and  the  millivoltmeter  which  reads  the  drop 
over  this  resistance.  In  this  case,  however,  the  other  two  de- 
flection instruments  must  read,  not  in  arbitrary  units,  but  in  volts 
or  millivolts. 

The  procedure  is  the  same  as  in  the  case  of  the  ring,  described 
above.  Giving  the  symbols  the  meanings  designated  in  Fig.  603, 
we  have,  with  the  key  K  open, 


With  the  key  K  closed,  we  have  by  readings  taken  simultaneously 
by  three  observers 

(2) 


E2  =  Cy.  (3) 

We  also  have  the  relation, 

C  =  Ci  +  /  =  Ci  +  g-  (4) 

From  Eqs.  (1),  (2),  (3)  and  (4)  we  deduce,  as  in  the  case  of  the 
measurement  of  the  resistance  of  a  ring, 


108  MEASURING  ELECTRICAL  RESISTANCE         [ART.  604 

,-M^a  (6) 

If  Eq.  (6)  is  used,  E\,  E2  and  V  can  be  multiplied  by  the  same 
constant  a  and  hence  the  deflection  instruments  may  be  cali- 
brated in  arbitrary  units,  provided  the  same  arbitrary  units  are 
used  for  all  three  instruments. 

604.  Measurement  of  the  Current  in  a  Bus-bar.  —  The  pur- 
pose to  be  fulfilled  in  finding  the  resistance  between  two  points 
in  a  bus-bar  is  to  enable  the  current  in  the  bus-bar  to  be  measured 
at  any  time  by  reading  the  drop  of  potential  between  the  points 
with  a  millivoltmeter.  A  portion  of  the  bus-bar  is  made  in  this 
manner  to  serve  as  a  shunt  for  a  millivoltmeter,  which  thus  becomes 

B 


1 

a                       R20 

b     \B 

M.V. 

rr\ 

FIG.  60 

an  ammeter  for  reading  the  current  in  the  bus-bar.  As  bus-bars 
are  made  of  copper  or  aluminum  which  have  a  large  temperature 
coefficient  we  have  to  consider  to  what  extent,  if  any,  their  change 
in  resistance  with  temperature  will  affect  the  precision  with  which 
the  current  may  be  read.  Let  Fig.  604  represent  an  arrangement 
to  be  employed. 

Here,  BBi  is  a  section  of  a  bus-bar.  We  shall  suppose  that 
the  resistance  R2o  has  been  accurately  obtained  at  20°  C.  between 
the  two  points  a  and  b  by  the  above  method.  The  millivolt- 
meter  M.V.  is  joined  to  the  points  a  and  b. 

Let  rT  =  r20  (1  +  aT)  (1) 

be  the  resistance  of  the  millivoltmeter  at  T°  C.  above  20°  C.  when 
r2o  is  its  resistance  at  20°  C.  and  a  is  the  temperature  coefficient 
of  its  winding. 

Let  Rt  =  #20  (1  +  00  (2) 

be  the  resistance  between  points  a  and  b  of  the  bus-bar  at  t°  C. 
above  20°  C.  when  R2o  is  its  resistance  at  20°  C.  and  0  is  the  tem- 
perature coefficient  of  the  material  of  the  bus-bar. 

The  bus-bar  may  change  in  temperature  both  from  changes  in 


ART.  604]     THE  MEASUREMENT  OF  LOW  RESISTANCE          109 

the  temperature  of  the  room  and  from  the  heating  due  to  the 
current  which  it  carries.  The  millivoltmeter  M.V.  can  only 
change  in  temperature  from  changes  in  the  temperature  of  the 
room.  Hence,  in  general,  the  temperature  of  the  millivoltmeter 
will  not  be  the  same  as  the  temperature  of  the  bus-bar. 

We  wish  to  determine  the  nature  and  magnitude  of  the  errors 
produced  by  these  temperature  changes  in  reading  the  current. 
If  I  is  the  current  in  the  bus-bar,  the  fall  of  potential  from  a  to 
b,  when  the  temperature  of  the  bus-bar  is  t,  will  be 

Et  =  IRt.  (3) 

The  current  thru  the  millivoltmeter  will  be 

C=^  =  KD,  (4) 

where  D  is  the  deflection  of  the  millivoltmeter  and  K  is  a  constant. 
Hence, 

Et  =  KDrT.  (5) 

By  Eqs.  (3)  and  (5) 


Since  the  bus-bar  and  the  winding  of  the  millivoltmeter  are 
both  of  pure  metal,  as  copper  or  aluminum,  the  temperature 
coefficients  a  and  /3  would  be  practically  the  same  and  may  be 
taken,  approximately,  0.004.  Eq.  (6)  can  therefore  be  written 

KDr2Q      1  +  0.004  T 
#20        1+  0.004  1   ' 

The  error  in  the  measurement  of  I  is  now  seen  to  depend  directly 
upon  the  amount  by  which  the  last  term  of  Eq.  (7)  departs  from 
unity.  In  the  case  of  no  heating,  by  the  current,  of  the  bus- 
bar above  room  temperature  (as  would  be  very  approximately 
realized  for  a  loading  of  the  bus-bar  of  50  per  cent  full  load  or 
less)  t  =  T,  and  there  is  no  error,  whatever  the  room  temper- 
ature becomes.  Now  t  can  never  be  less  than  T  but  may  assume 
a  value  T  -\-  dT  where  8T  represents  the  temperature  of  the  bus- 
bar, above  the  temperature  of  the  air.  In  this  case  Eq.  (7)  be- 

comes 

KDr20  1  +  0.004  T 

#20         1+  0.004  T  +  0.004  dT' 


110  MEASURING  ELECTRICAL  RESISTANCE  [ART.  605 

As  a  rather  extreme  case  we  may  take  T  =  10°  C.  above  20°  C., 
and  8T  =  5°  C.  Then 

1  +  0.004  X  10  =  L04  = 

1  +  0.004  X  10  +  0.004  X  5      1.06 

Thus  the  true  value  of  the  current  would  be,  in  this  case,  about 
2  per  cent  less  than  one  would  read  it  upon  the  millivoltmeter. 

The  following  estimate  shows  that  the  fall  of  potential  in  a 
bus-bar  is  large  enough  to  apply  the  above  method  for  measur- 
ing the  current  in  it. 

The  resistance  of  100  per  cent  conductivity  copper  at  20°  C. 
is  67.7  X  10~8  ohm  per  linear  inch  per  square  inch  of  cross-section. 
It  is  good  practice  to  allow  1000  amperes  per  square  inch  of  cross- 
section  of  copper  conductor.  Then,  with  1000  amperes  to  the 
square  inch  of  cross-section,  the  drop  of  potential  per  linear  inch 
becomes 

103  X  67.7  X  10~8  =  0.677  X  10~3  volt, 

or  0.677  millivolt  per  linear  inch.  If  the  full  scale  reading  of  the 
millivoltmeter  is  20  millivolts,  the  distance  between  the  potential 
points,  a  and  b  (Fig.  604),  would  need  to  be 

=  29.5+  inches. 


This  length  of  bus-bar,  to  be  used  for  the  purpose  of  a  shunt, 
could  be  obtained  behind  almost  any  switch  board,  and  it  is  prob- 
able that  a  shunt  for  the  millivoltmeter  of  this  character  would 
serve  quite  as  well  and  perhaps  be  superior  to  the  shunts  ordinarily 
used.  For,  these  latter  have  a  very  low  temperature  coefficient 
and  changes  in  the  temperature  of  the  room  will  increase  the 
resistance  of  the  millivoltmeter  without  increasing  in  like  degree 
the  resistance  of  the  shunt,  and  hence  there  is  no  automatic  com- 
pensation, as  in  the  case  discussed  above,  where  the  bus-bar  itself 
serves  as  a  shunt. 

To  make  the  millivoltmeter  read  directly  in  amperes  requires, 
of  course,  that  the  constant  K  in  Eq.  (8)  be  correctly  chosen. 
As  we  are  at  liberty  to  give  any  value  to  the  resistance  r2o  it  will 
always  be  possible  to  do  this. 

605.  Measurement  of  the  Resistance  of  Underground  Mains. 
—  An  important  application  of  the  methods  described  above  for 
measuring  the  resistance  of  a  portion  of  a  closed  circuit  is  the 
determination  of  the  resistance  between  two  selected  potential 


ART.  605]      THE   MEASUREMENT  OF  LOW  RESISTANCE 


111 


points  upon  an  underground  gas  or  water  main.  Underground 
pipes  are  subject  to  deterioration  from  electrolysis,  caused  by 
"  tramp  "  currents  which  get  into  the  pipe  line  from  neighboring 
electric  trolley  roads.  The  electrolysis  occurs  when  current  leaves 
the  pipes.  It  becomes  important,  at  times,  to  be  able  to  quickly 
and  accurately  measure  the  current  which  flows  in  some  selected 
section  of  a  pipe  line.  It  is  evident  that  this  can  be  easily  accom- 
plished by  measuring,  with  a  millivoltmeter,  the  potential  drop 
between  two  points  on  a  section  of  pipe,  provided  the  resistance 
between  these  two  points  has  been  previously  determined.  The 
method  given  in  Fig.  605a,  which  is  a  slight  modification  of  those 


FIG.  605a. 

described  above,  enables  this  resistance  to  be  measured  with 
considerable  precision  while  the  section  of  pipe  is  in  place  in  the 
pipe  line. 

The  measurement  is  made  with  two  millivoltmeters  and  an  am- 
meter. One  or  two  cells  of  storage  battery  are  also  required. 
The  cells  of  storage  battery,  a  key  K  and  the  ammeter  A  are 
joined  in  series  and  connected  at  two  places,  as  shown  in  the  dia- 
gram, to  a  section  of  pipe.  These  connections  are  best  made  by 
drilling  0.25-inch  holes,  about  half  way  thru  the  pipewall,  and 
driving  brass  plugs  into  them.  Heavy  copper-wire  connections 
may  then  be  soldered  to  the  brass  plugs.  The  other  connections, 
which  serve  as  potential  points,  may  be  made  in  a  similar  manner, 
but  smaller  holes  and  plugs  will  serve.  There  should  be  as  much 
separation  as  possible  between  a  potential  point  and  the  place  of 
connection  of  a  current  lead,  and  these  should,  preferably,  be 
located  at  the  ends  of  diameters  of  the  pipe  which  form  with 
each  other  an  angle  of  90  degrees.  It  is  well  to  take  one  set  of 
readings  and  calculate  the  resistance  with  the  polarity  of  the 
storage  cell  in  one  direction,  and  then  take  a  second  set  with  the 
polarity  of  this  cell  reversed.  By  taking  the  mean  of  the  two 


112  MEASURING  ELECTRICAL  RESISTANCE         [ART.  605 

resistances,  thus  obtained,  the  error  is  largely  eliminated  which 
results  from  the  flow  lines  of  current  from  the  storage  cell  not 
being  parallel  with  the  section  of  pipe  between  the  potential 
points.  This  is  specially  the  case  when  there  is  considerable  cur- 
rent flowing  in  the  pipe  from  other  sources  than  the  storage  cell. 
This  error  will  be  small,  however,  in  any  case,  if  the  distance  be- 
tween a  potential  point  and  the  point  of  connection  of  a  current 
lead  is,  say,  twice  the  diameter  of  the  pipe  and  these  terminals 
are  located  as  above  suggested. 

Calling  I  the  current  thru  the  ammeter  and  referring  to  Fig.  605a 
for  the  meaning  of  the  other  symbols  we  have,  as  in  the  cases 
given  above,  with  the  key  K  open 

;-£-»•  » 

and  with  the  key  K  closed 

El  =  dx,  (2) 

#2  -  Cy,  (3) 

and 

C  =  Ci  +  7, '  (4) 

from  which  we  find 


-y-  (s) 

Also 

1 

V  =  -  -J-^-'  (6) 

or 


In  applying  the  method,  one  is  not  in  the  least  troubled  by  the 
sudden  variations  of  the  current  in  the  pipe  which  constantly 
occur,  because  EI  and  E%'  are  read  simultaneously  to  obtain  the 
ratio  N  and  then,  again,  EI,  E2,  and  the  ammeter  A  are  read 
simultaneously  to  obtain  the  other  necessary  values.  Three 
observers,  reading  at  the  same  moment,  obtain  correct  values;  for, 
when  the  current  varies,  a  variation  occurs  in  all  three  instruments 
at  the  same  time,  the  proper  relation  between  the  readings  of  the 
three  instruments  being  always  maintained. 

This  method  was  carefully  tested  by  the  author  upon  an  actual 
pipe  line  with  excellent  results.  The  essential  features  of  the 
test  are  recorded  below: 


ART.  605]      THE   MEASUREMENT  OF  LOW  RESISTANCE 


113 


The  diameter  of  the  pipe  was  15  inches.  Two  pipe  lengths  were 
uncovered  and  the  connections  to  the  pipe  sections  were  made 
at  distances  and  in  the  manner  shown  in  Fig.  605b. 

The  method  embodied  the  use  of  two  cells  of  storage  battery, 
which  would  yield  on  short  circuit,  when  joined  in  parallel,  from 
125  to  150  amperes,  also  one  ammeter  reading  to  200  amperes  and 
two  millivoltmeters  giving  a  full  scale  deflection  with  20  millivolts. 
In  circuit  with  the  ammeter  and  storage  cell  a  single  pole  current 
switch  K  was  used.  The  following  readings  were  taken: 


Readings  to  Left=  + 


Formula  Assumes 
Current  Flows  thus 
1 


West     k.958^i  j. 7.916---*]  j*^  k.884^  i- 7.916' H  U-^,  „  /  East 

1 094  ^NUWB 

FIG.  605b. 

Ei  and  Ez'  were  read  simultaneously.  The  current  in  the  pipe 
was  sufficient  for  the  purpose.  The  mean  of  nine  readings  of 
EI  was  7.511  millivolts  and  the  mean  of  nine  readings  of  E2f  was 


7.122  millivolts. 


Hence  the  value  of  the  ratio  of  -  was 

y 


N  = 


7.511 
7.122 


=  1.055. 


There  were  then  taken  seventeen  sets  of  readings  of  EI,  E2  and 
I  with  the  positive  pole  of  the  battery  cell  joined  to  No.  7  terminal 
and  a  like  number  with  the  negative  pole  joined  to  this  terminal. 
The  following  table  exhibits  a  few  sample  readings: 


No.  1  current  -+-  to  No.  7  terminal. 

No.  2  current  —  to  No.  7  terminal. 

ft 

M.  V. 

M.V. 

/  amperes 

x  =  ohm 
for  7.916  ft. 

E^ 
M.V. 

M.V. 

7  amperes 

x  =  ohm 
for  7.916  ft. 

-68 
-6.2 
-8  3 

1.6 

1.8 
0.9 

116 
110 
127 

0.0000731 
0.0000736 
0.0000728 

7  60 

7.80 
8.20 

04 
08 
1.0 

-102 

-  98 
-101 

0.0000703 
0.0000709 
0.0000707 

The  mean  value  deduced  for  x  with  the  current  from  the  storage 
cell  positive  to  terminal  No.  7  was  0.00007315  ohm,  and  with  the 
current  from  the  storage  cell  negative  to  terminal  No.  7  was 


114  MEASURING  ELECTRICAL  RESISTANCE         [ART.  606 

0.00007077  ohm.  The  mean  of  these  two  results  is  0.00007196 
ohm  for  a  length  of  the  pipe  of  7.916  feet.  There  were  40  feet 
of  No.  14  wire  used,  as  potential  leads,  to  each  millivoltmeter. 
Calculation  showed  that  to  correct  for  the  resistance  of  these  leads 
the  final  value  of  x  should  be  multiplied  by  1.088.  Doing  this 
and  reducing  the  resistance  to  a  length  of  one  foot  of  pipe,  the 
final  value  found  was  9. 91  microhms  per  foot,  at  65°  F. 

The  test  was  defective  in  that  the  distances  between  potential 
points  and  points  of  attachment  of  current  leads  were  not  chosen 
as  great  as  they  should  have  been  and  were  all  made  on  the  top 
side  of  the  pipe.  The  "  tramp  "  currents  in  the  pipe  were  large 
and  very  variable  at  the  time  of  the  test.  In  spite  of  this  the 
resistance  measurement  is  probably  correct  within  1.5  or  2  per  cent, 
and  should  have  been  better. 

It  was  found  in  this  test  that  care  had  to  be  exercised  to  give 
to  the  readings  of  the  three  instruments  the  proper  algebraic  signs. 
By  making  a  diagram,  like  Fig.  605b,  before  beginning  the  test, 
errors  of  this  character  may  be  avoided,  which  otherwise  might 
occur  and  entirely  vitiate  the  results. 

606.  Comparison  of  Low  Resistances  by  the  Modified  Slide- 
wire   Bridge.  —  The  theory  of  this  method   has  already   been 
given,  par.   402,  to  which   reference  should   be  made.     As  the 
method  requires  two  settings  of  a  sliding  contact  and  a  balance 
twice  obtained,  it  is  inferior  to  the  Kelvin-double-bridge  method 
(§  609).     It  would  be  used  generally  only  when  the  necessary 
resistance  units,  for  building  up  a  Kelvin  double  bridge,  are  not 
obtainable. 

607.  Comparison  of  Low  Resistances  by  the  Carey-Foster- 
Bridge   Method.  —  The  theory  and  application  of  the  Carey- 
Foster  bridge  has  been  given  in  par.  403. 

When  a  low  resistance  is  provided  with  potential  points  and 
leads,  it  is  not  adapted  for  use  as  a  standard  with  which  to  com- 
pare another  resistance  by  the  Carey-Foster-bridge  method.  This 
method  is  confined  to  the  measurement  of  such  low  resistances 
as  are  provided  with  terminals  which  may  be  inserted  into  mercury 
cups  or  clamped  in  binding  posts.  Thus  the  Carey-Foster  method 
is  hardly  to  be  considered  a  method  of  low-resistance  measure- 
ment. By  the  use  of  connecting  leads,  however,  for  the  standard 
resistance  and  the  resistance  under  comparison,  which  are  alike 
in  size,  resistance  and  temperature  coefficient  the  comparison  can 


ART.  609]      THE   MEASUREMENT   OF  LOW   RESISTANCE  115 

be  made  without  any  error  being  introduced  due  to  lead  resist- 
ances. The  method  is  one  admirably  adapted  to  determinations 
of  temperature  coefficients  of  comparatively  short  lengths  of  low- 
resistance  wires.  In  temperature-coefficient  determinations  the 
absolute  values  of  the  resistances  are  of  less  importance  than  the 
variations  with  temperature  of  these  resistances,  and  the  Carey- 
Foster  method  will  give  these  with  great  precision. 

608.  Comparison  of  Low  Resistances  with  a  Potentiometer. — 
Two  low  resistances  which  are  provided  with  potential  terminals 
may  be  compared  with  considerable  precision  by  using  a  poten- 
tiometer.    The  two  resistances  are  joined  in  series  and  current 
is  passed  thru  them  from  a  source  of  constant  E.M.F.  as  a  storage 
cell  of  large  capacity.     The  fall  of  potential  is  read  with  the 
potentiometer,  in  quick  succession,  first  over  the  one  resistance 
and  then  over  the  other.     The  resistances  are  in  the  same  ratio 
as  the  potential  drops,  provided  the  current  thru  the  resistances 
remains  constant  during  the  taking  of  the  two  readings. 

In  using  a  potentiometer  in  this  way  it  is  unnecessary  to  make 
use  of  a  standard  cell  as  the  absolute  values  of  the  drops  in  potential 
are  not  required.  As  a  good  potentiometer  will  compare  E.M.F.'s 
as  low  as  0.01  volt,  to  accuracies  of  the  order  of  0.1  of  1  per  cent, 
it  is  not  necessary  to  pass  a  very  large  current  thru  the  two  low 
resistances  under  comparison,  and  hence  the  comparatively  small 
current  needed  -can  be  maintained  very  constant  for  the  short  time 
required. 

609.  The  Kelvin  Double  Bridge;   A  Network  of  Nine  Con- 
ductors. —  The  most  elegant  and  the  most  precise  methods  for 
comparing  any  two  resistances,  which  are  provided  with  potential 
points,  are  those  which  employ  some  form  of  the  Kelvin  double 
bridge.     We  shall,  therefore,  discuss  fully  the  theory,  the  forms, 
and  the  applications  of  this  unique  arrangement  of  circuits  for  the 
measurement  of  resistances. 

The  Kelvin  double  bridge,  due  to  Lord  Kelvin,  is  a  network 
of  nine  conductors.  This  network  may  be  conveniently  repre- 
sented (P  and  Q,  Fig.  609a)  by  either  of  the  two  following  diagrams, 
which  are  exactly  equivalent  electrically.  The  diagrams  are 
drawn  so  that  like  letters  refer  to  like  branches  of  the  network. 

Referring  to  the  diagram  (P),  we  note  that  the  Kelvin  net-work 
becomes  a  Wheatstone  net-work,  when  d  equals  zero  and  also  when 
d  equals  infinity. 


116 


MEASURING  ELECTRICAL  RESISTANCE        [ART.  609 


When  d  equals  zero  the  net-work  assumes  the  form  shown  at 
(p),  Fig.  609b.     Here  a  and  /3  become  two  resistances  in  parallel, 


FIG.  609a. 

which  are  in  circuit  with  the  galvanometer,  and  hence  do  not 
enter  into  the  equation  of  the  bridge,  which  in  this  case  is 

Ab  -  Ba  =  0.  (1) 

When  d  equals  infinity  the  net-work  assumes  the  form  shown  at 
(q).     Here  a  and  /3  are  thrown,  the  one  in  series  with  A  and  the 


FIG.  609b. 

other  in  series  with  B.     Thus  the  equation  of  the  bridge  in  this 
case  becomes, 

(A  +  a)  b  -  (B  +  0)  a  =  0.  (2) 

In  the  actual  employment  of  the  Kelvin  double  bridge  the  resist- 


ART.  610]     THE  MEASUREMENT  OF  LOW  RESISTANCE          117 

ance  d  can  never  be  made  zero,  -tho  it  should  be  made  as  small  as 
possible. 

Referring  to  Fig.  609a,  A  and  B  are  two  low  resistances  (pro- 
vided with  potential  points  at  1,  2,  and  1',  2')  which  are  to  be 
compared,  the  resistance  of  both  A  and  B  being  that  included 
between  potential  points.  The  resistance  d,  called  the  "  yoke," 
connects  A  and  B.  Besides  being  made  as  low  as  possible  it  is 
arranged  to  be  easily  removed  as,  for  example,  by  withdrawing  a 
copper  link  from  two  mercury  cups.  The  other  four  resistances  a, 
b  and  a,  /3  are  relatively  high  resistances  and  are  called  the  ratio 
coils.  Because  there  are  two  pairs  of  ratio  coils  the  net-work  is 
called  a  "  double  bridge." 

610.  Theory  of  the  Kelvin  Double  Bridge.  —  The  relations 
which  must  connect  the  resistances  to  have  the  bridge  balanced, 
that  is,  to  have  no  current  flowing  thru  the  galvanometer,  may  be 
found  as  follows:  Referring  to  diagram  (P)  of  Fig.  609a  we  see 
that,  when  there  is  no  current  in  the  galvanometer,  the  current 
in  the  branch  a  must  be  the  same  as  the  current  in  the  branch  6. 
Call  this  current  ii.  Also  the  current  in  the  branch  a  must  be 
the  same  as  the  current  in  the  branch  j8.  Call  this  current  i. 
Also  the  current  in  the  branch  A  will  equal  the  current  in  the 
branch  B.  Call  this  current  7.  By  the  law  of  branched  circuits, 


Further,  the  fall  of  potential  from  the  point  1  to  the  point  3  must 
be  the  same  as  the  fall  of  potential  from  the  point  1  to  the  point  4. 
Otherwise  there  would  be  a  difference  of  potential  between  points 
3  and  4  and  the  galvanometer  would  deflect.  Call  this  fall  of 
potential  Ea.  In  like  manner,  the  fall  of  potential  from  point  3 
to  point  2'  will  be  the  same  as  the  fall  of  potential  from  point  4  to 
point  2'.  Call  this  fall  of  potential  E&.  Now, 

Ea  =  I  A  +  ia  =  ha,  (2) 

and 

Eb  =  IB+  iff  =  ij>.  (3) 

Hence  taking  the  ratio  of  Eqs.  (2)  and  (3)  we  find 

a      I  A  +  ia  m 

b      IB+  i0  ' 


118  MEASURING  ELECTRICAL  RESISTANCE         [ART.  610 

Substituting  in  Eq.  (4)  the  value  of  i  given  in  Eq.  (1)  we  have, 

a      A(a+ft  +  d)+da 
b       B  ' 


Eq.  (5)  is  readily  put  in  the  form, 

A_  _  a       d  ft          la  _  a 

In  Eq.  (6)  call, 


B    a  +  ft 
and  we  have 


(7) 


The  quantity  A;  is  a  correction  factor  which  must  be  either  cal- 
culated and  added  to  the  ratio  T  to  give  the  ratio  A  to  #,  or  it 

must  be  made  so  small  that  it  may  be  neglected  even  for  work  of 
the  highest  precision.  In  practice  the  latter  course  is  followed. 
We  have  to  consider  how  k  may  be  reduced,  and  also  the  magnitude 
of  the  errors  which  will  be  introduced  by  neglecting  k  when  it  is 
not  absolutely  zero. 

The  correction  factor  k  may  be  considered  as  the  product  of 
three  parts, 

d  ft  ,     a      a 

B'  a  +  0  +  d     and     6  ~  -ft 

Each  of  these  parts  should  be  made  as  small  as  possible.  The 
yoke  d  should  be  made  very  small  in  resistance  tho  this  cannot 
be  made  absolutely  zero.  This  means  that  the  resistances  A  and 
B  should  be  joined  together  by  a  heavy  copper  conductor  which 
makes  as  good  contact  as  possible  with  A  and  B.  The  smaller 
the  resistance  B,  the  more  important  it  becomes,  for  keeping  the 

ratio  -g  small,  to  make  d  as  small  as  possible.     If  B  is  over  0.01 

ohm,  -5  may  be  made  fairly  small. 
n 

/Q 

The  part  —        —  -  will  always  be  less  than  unity.     As  d  is 

a  +  ft  -\-  a 

negligible  the  value  of  this  part  cannot  be  varied,  because  the 
relative  values  of  a  and  ft  are  determined  by  the  relative  values 
of  the  resistances  A  and  B,  being  compared.  Thus,  if  a  is  multi- 


ART.  610]      THE   MEASUREMENT   OF  LOW   RESISTANCE  119 

plied  by  any  factor,  0  must  be  multiplied  by  the  same  factor,  and 
the  magnitude  of  the  fraction  remains  unchanged. 

The  third  part  ?  —  ^  is  the  most  important  one,  for  by  care 
o      j8 

in  making  adjustments  it  may  be  made  as  small  as  we  please. 
It  becomes  zero,  as  does  also  k,  when,  accurately,  -r  =  -• 

An  estimation  of  the  magnitude  of  the  errors  resulting  from  a 
neglect  of  the  correction  factor  k  has  been  made  in  a  pamphlet 
issued  by  The  Leeds  and  Northrup  Company,  of  Philadelphia, 
and  is  here  reproduced  in  substance. 

If  a  and  b,  a  and  j3  are  resistances  adjusted  to  be  like  each  other 
to  0.01  of  1  per  cent  and  are  arranged  so  that,  according  to  their 

nominal  values  7  =  7-  then,  the  greatest  value  which  r  —  7-  can 
op  op 

have  is  0.00020;  the  probable  amount  is  less  than  this.  This 
value  is  based  on  the  assumption  that  the  inequalities  of  0.01  of 
1  per  cent  group  themselves  in  such  a  way  as  to  form  the  maximum 

fj  R 

error.     Assuming  ^  =  1  and       ,    Q   .    ,  =  1  (which  is  obviously 
£>  a  ~T  p  -\-  d 

more  than  it  ever  will  be)  the  complete  formula  becomes  in  this 
case  4  =  ?  +  0.00020,  which  may  be  written  A  =  B  fe  +  0.00020^  • 

If  r-  =  1,  as  will  be  the  case  when  two  resistances  of  equal  value 
b 

are  under  comparison,  the  maximum  error  in  this  case  will  be 
0.02  of  1  per  cent.  If  r  =  10,  as  will  be  the  case  when  the  known 
resistance  is  ten  times  as  large  as  the  unknown,  the  maximum  error 
will  be  0.002  of  1  per  cent.  If  r  =  0.1,  as  will  be  the  case  when  the 

known  resistance  is  0.1  of  the  unknown,  the  maximum  error  will 
be  0.2  of  1  per  cent.  These  considerations  show  that  for  ordinary 
cases  where  the  connecting  resistance  can  be  kept  smaller  than 
the  unknown  and  the  unknown  resistance  is  equal  to  or  smaller 
than  the  known,  that  no  attention  need  be  paid  to  the  correction 
term  provided  the  ratio  coils  are  like  each  other  to  0.01  of  1  per 
cent. 

There  may  be  cases  in  which  it  is  not  possible  to  meet  all  these 
conditions. 


120 


MEASURING  ELECTRICAL  RESISTANCE         [ART.  611 


In  these  cases  the  following  procedure  is  usually  possible.  Re- 
ferring again  to  Fig.  609a  it  is  evident  that  if  the  connector  d  is 
removed,  the  combination  forms  an  ordinary  Wheatstone  bridge  in 


which  for  a  balance  we  must  have  r 


A  +  a 


If  the  galvanom- 


a  or 


eter  does  not  show  a  balance  it  may  be  brought  about  by  shunting 
When  j-  is  made  exactly  equal  to  p   ,   *?  then  indeed  T 

will  be  very  nearly  equal  to  ^  because  A  and  B  will  always  be  very 

p 
low  resistances  compared  with  a  and  /3  and,  furthermore,  they  are 

in  the  same  ratio  as  a  and  /3.    Consequently  when  T  is  made  equal 

"  it  may  generally  be  assumed  without  error  that  r  =  -  • 

op 


n 
B 


611.  Sensibility  Which  Can  be  Obtained  With  the  Kelvin 
Double  Bridge.  —  We  shall  now  examine  the  percentage  varia- 
tion in  one  of  the  low  resistances  which  can  be  detected  when 


MA, 


A    I 


I, 


I 


I-dl 
d 


V  B  * 

FIG.  611. 

using  a  galvanometer  purchasable  upon  the  market.  An  inquiry 
involving  a  rigid  solution  of  the  problem  would  require  a  compli- 
cated application  of  Kirchoff's  laws.  But  this  is  unnecessary  as  a 
very  closely  approximate  solution  may  be  obtained  quite  simply 
as  follows: 

Represent  the  bridge  as  in  Fig.  611.  First,  let  the  bridge  be 
balanced  when  contact  p  is  at  the  point  1  on  A.  Let  the  current 
in  A  be  7.  It  will  then  be  7  —  67  in  d.  But  67  will  be  a  very 
small  quantity  as  compared  with  7,  and,  further,  the  value  of 
7  will  be  very  little  affected  by  small  displacements  of  the  contact 
p  upon  A.  We  may,  therefore,  without  sensible  error  consider 
the  current  at  all  times  to  be  7  in  the  branches  A ,  d,  B,  and  in  the 
battery  circuit.  Now  displace  p  to  point  li  thus  increasing  A  by 


ART.  611]     THE  MEASUREMENT  OF  LOW  RESISTANCE          121 

a  small  quantity  dA.  The  fall  of  potential  from  1  to  2  was  I  A 
before  the  displacement,  and  this  fall  of  potential  was  required  to 
balance  the  bridge.  The  fall  of  potential  from  li  to  2  is  I  A  -+-I5A. 
Hence,  increasing  the  fall  of  potential  by  the  amount  IdA  is 
equivalent  to  introducing  in  circuit  with  A  a  small  E.M.F.  This 
E.M.F.  sends  a  current  thru  the  galvanometer.  This  current  must 
pass  thru  the  two  resistances  a  and  a  in  series  with  the  gal- 
vanometer, and  thru  the  galvanometer  resistance  g  shunted  by 
the  resistances  6,  B  and  /3  in  series. 

We  may  assume  that  this  E.M.F.  sends  no  current  thru  d  or 
the  battery  because  we  have  considered  that  /  remains  constant. 
Also  the  resistances  A  and  B  do  not  need  to  be  considered  as  they 
are  very  small  compared  with  the  others.  For  the  current  which 
flows  thru  a,  due  to  the  E.M.F.  IdA,  we  thus  obtain, 


(1) 

where  B.a  +  a+l£+0L. 

The  current  thru  the  galvanometer  will  be 


From  Eqs.  (1)  and  (2)  we  find 

c  __  ISA(b  +  0) 

°"      (a  +  «)  (0  +  6  +  0)  +0(6  +  0) 

If  do  =  KC0)  where  do  denotes  the  deflection  of  the  galvanometer 
in  scale  divisions  when  the  scale  is  1000  scale  divisions  from  the 
mirror,  and  K  is  the  galvanometer  constant,  we  have 
do_        _  8A  (b  +  0) 
KI      (a  +  a)(g  +  b+P)+g(b  +  ft)' 

In  ordinary  practice  we  would  have  a  =  a  and  b  =  /3. 

Give  these  values  to  a  and  #.     We  then  find,  in  solving  Eq.  (4) 

for  dA,  dividing  it  by  A,  and  multiplying  it  by  100  that, 


Eq.  (5)  expresses  the  percentage  variation  in  A  which  will  produce 
a  deflection  d$  in  the  galvanometer,  the  battery  current  being 
I  amperes.  The  author  has  verified  Eq.  (5)  experimentally. 


122  MEASURING  ELECTRICAL  RESISTANCE        [ART.  611 

We  proceed  to  apply  Eq.  (5)  to  calculations  of  the  percentage 
precision  with  which  certain  low  resistances  may  be  compared  by 
the  Kelvin-double-bridge  method.  First  take  the  following  case: 

Let  d0  =  10"1  of  a  division  as  the  smallest  departure  from  zero 
which  can  be  detected  with  certainty. 

Let  A  =  B  =  10-3  ohm. 

Let  7  =  1  ampere. 

Let  a  =  102  ohms,  then  b  =  102  ohms. 

Let   g  =  180  ohms. 

Let  K  =  1.21  X  108.* 

Substituting  these  values  in  Eq.  (5)  we  obtain 
A  A  1  O"1  V  1  02 

A  10°  -  1.21  xVx  IX  10-*  X  (200  +180X2)  . 

=  1  21       1Q4  X  560  =  0.046  +  per  cent. 

Second,  take  the  case  in  which  A  =  10~3  ohm,  and  B  =  10~2 
ohm.  Then  if  a  =  102  ohms,  we  shall  have  b  =  103  ohms. 

If  all  the  other  quantities  are  the  same  as  in  the  first  case,  we 
find 


21  x  1Q4 

Third,  take  the  case  in  which  A  =  10~2  ohm  and  B  =  10~3  ohm. 
Then  if  a  =  103  ohms,  we  shall  have  b  =  102  ohms,  and  if  all  the 
other  quantities  remain  the  same 

*£  100  =  —  ^-    -^  X  3980  =  0.0328  +  per  cent. 
^4.  1.^1  X  1U 

We  thus  note  from  the  second  and  third  cases  that  the  precision  is 
the  same  whether  we  make  A  =  0.001  ohm  and  B  =  0.01  ohm 
or  A  =  0.01  ohm  and  B  =  0.001  ohm.  In  the  above  examples 

7=1  ampere.     But  the  per  cent  displacement  -r-  100  diminishes 

A. 

*  The  values  of  g  and  K  are  those  belonging  to  a  galvanometer  made  by 
the  Leeds  and  Northrup  Co.,  of  Philadelphia,  known  as  their  No.  2280,  narrow- 
coil  galvanometer.  This  galvanometer  has  a  complete  period  of  1.8  seconds. 
Its  figure  of  merit  is 

FOTO  121.8  _    ...    -,-^,   . 

=  -  -=.  =  _  .     .  _  =.2.8  D'Arsons. 
T72  VR      DPV180 

See  "The  Comparison  of  Galvanometers  and  a  New  Type  of  Flat-coil  Gal- 
vanometer," by  E.  F.  Northrup,  Jour,  of  the  Franklin  Inst.,  Oct.,  1910.  See 
also  par.  1504,  and  the  table. 


ART.  612]      THE   MEASUREMENT   OF  LOW  RESISTANCE  123 

directly  as  /  increases,  that  is,  the  possible  sensibility  in  the 
measurement  increases  as  /  increases.  A  resistance  as  low  as 
0.01  ohm  may  generally  be  made  to  carry  a  current  as  high  as 
20  or  more  amperes  without  undue  heating.  Hence,  by  using  a 
current  of  20  amperes  the  sensibility  in  the  above  three  cases 

would  be  increased  20  times,  that  is,  100  —r  would  be  0.0023  of  1 

A. 

per  cent  in  the  first  case  and  0.00164  of  1  per  cent  in  the  other 
two  cases.  These  results  show  that  resistances  of  the  order  of 
0.001  ohm  may  be  compared  by  the  Kelvin  double  bridge  with 
much  the  same  order  of  accuracy,  as  resistances  of  the  order  of 
1000  ohms  may  be  compared  by  the  Wheatstone  bridge. 

612.  Methods  of  Applying  the  Kelvin-Double-Bridge  Prin- 
ciple. —  From  the  formula  of  the  Kelvin  double  bridge  A  =  =-  B 

when  T  =  -  by  construction,  it  is  seen  that,  if  B  is  the  standard 
o      p 

resistance,  a  balance  may  be  obtained  either  by  varying  B  or  by 
varying  the  ratio  r ,  provided  the  ratio  —  is  varied  also  in  the  same 
way.  A  combination  of  these  two  methods  for  securing  a  balance 

may  likewise  be  applied.     In  this  case  the  ratios  -r  and  ^  would  be 

o         P 

given  values  such  as  10,  1,  0.1,  etc.,  and  the  standard  B  would  then 
be  varied  in  infinitesimal  steps  to  secure  an  exact  balance. 

The  method  whereby  B  is  maintained  fixed  and  the  ratios 

r  and  -  are  similarly  varied  to  secure  a  balance  is  that  adopted  by 

the  instrument  maker,  Otto  Wolff  of  Berlin.  For  accomplishing 
this  end  a  special  box  of  ratio  coils  is  provided.  This  box  has 
four  double  dials  which  give  the  values  of  a  and  of  a  in  steps  of 
0.1  ohm  and  two  sets  of  blocks  with  plugs  with  which  the  values 
of  b  and  p  are  set  at  25,  50,  100,  or  25  +  50,  25  +  100,  50  +  100, 
or  25  +  50  +  100.  The  general  plan  of  the  connections  is  given 
below  in  Fig.  612a. 

Here  the  dials  aioo,  «io,  ai,  «0.i  and  «ioo,  «io,  «i,  «o.i  are  varied  at 
the  same  time  by  turning  a  single  handle  for  each  pair  of  dials. 
The  dial  a^o  reads  in  ten  steps  of  100  ohms  each,  the  dial  ai0  in 
ten  steps  of  10  ohms  each,  the  dial  a\  in  ten  steps  of  1  ohm  each 
and  the  dial  a0.i  in  ten  steps  of  0.1  ohm  each. 


124 


MEASURING   ELECTRICAL   RESISTANCE         [ART.  612 


The  resistances  b  and  (3  have  their  values  varied  by  plugging 
between  brass  blocks,  between  which  there  are  resistance  coils. 
The  value  obtained  for  b  and  0  is  that  of  the  coils  between  the 
blocks  left  unplugged.  The  points  1  and  2  are  joined  to  the 
potential  points  of  the  resistance  to  be  measured  and  the  points 


FIG.  612a. 


18  and  2a  are  joined  to  the  potential  points  of  the  standard.  To 
go  with  a  box,  such  as  the  above,  fixed  standards  of  low  resistance, 
furnished  with  potential  points,  are  required.  A  typical  set  of 
standards,  such  as  are  furnished  by  Otto  Wolff,  would  be: 


Nominal  value 

Load  capacity  (approxi- 
mate) 

Ohm 

1 

Watts 
1 

0.1 

5 

0.01 

500 

0.001 

500 

0.001 

1000 

0.0001 

1000 

0.0001 

2500 

0.00001 

2500 

The  above  standard  resistances  of  the  lower  values  are  arranged 
to  be  water  cooled.  When  used  as  standards  for  measuring  current 
by  reading  the  fall  of  potential  with  a  potentiometer  between  their 
potential  points  they  may  be  much  more  heavily  loaded  than 
when  used  as  precision  standards  for  comparison  resistances.  As 
they  are  made  of  manganin  their  temperature  coefficient  is  small, 
being  of  the  order  0.00002  per  ohm  per  degree  centigrade. 


ART.  612]      THE   MEASUREMENT   OF  LOW  RESISTANCE 


125 


The  method  adopted,  by  leading  American  instrument  makers, 
for  the  application  of  the  Kelvin  double  bridge  is  the  mixed  method. 

In  this  the  ratios  ?  and  ^  are  set  to  various  values,  and  the  positions 
b         ft 

of  two  contacts,  made,  one  with  a  plug  and  one  by  means  of  a 
knife-edge  sliding  upon  a  bar  of  manganin,  are  varied  until  a 
balance  of  the  bridge  is  obtained. 
This  method  is  diagrammatically  shown  in  Fig.  612b. 


FIG.  612b. 


A  comparison  of  Fig.  612b  and  Fig.  609a  will  make  the  relation 
of  the  parts  clear.  A  is  a  bar  of  manganin  which,  in  one  con- 
struction, has  a  resistance  of  0.001  ohm  between  the  points  0  and 
100  of  a  vernier  scale  which  is  placed  alongside  the  bar.  It  will 
carry  150  amperes.  Upon  this  bar  slides  a  knife-edge  contact. 
The  left  end  of  the  bar  joins  to  a  heavy  strip  of  manganin.  At 
points  separated  by  a  resistance  exactly  equal  to  that  of  the  bar, 
namely  0.001  ohm  in  the  case  cited,  are  brought  off  9  potential 
leads  to  brass  blocks.  An  extra  potential  lead  is  brought  off  from 
the  bar  opposite  the  0  of  the  scale.  The  two  contacts  I  and  m 
correspond  to  the  two  contacts  1  and  2  of  Fig.  609a.  The  con- 
tact I  is  made  with  the  block,  to  which  a  potential  lead  is  joined, 
by  means  of  a  plug.  Altogether  a  resistance  of  0.01  ohm,  in 
infinitesimal  steps,  may  be  obtained  between  the  contacts  I  and 
m.  B  represents  the  unknown  resistance  to  be  measured  between 
the  potential  points  n  and  0.  The  two  sets  of  ratio  coils,  which 
in  one  construction  are  mounted  together  in  a  separate  box,  are 
shown  at  a,  b  and  a,  j8.  Many  different  ratios  may  be  obtained  by 


126  MEASURING  ELECTRICAL  RESISTANCE         [ART.  613 

suitably  plugging  this  box.  For  example,  the  ratio  value  given  in 
the  diagram  is 

•    b      ft        100  D      b  . 

a  =  a=  10000-     Hence,  as  B  =- A, 

we  would  obtain  a  balance,  if  B  =  0.00001831,  when  A  =  0.001831, 
which  value  may  be  read  from  the  setting  of  the  plug  at  I  and  by 
the  vernier  of  the  scale  set  at  m. 

Excellent  apparatus  for  the  above  adaptation  of  the  Kelvin 
double  bridge  to  low-resistance  measurements  has  been  placed 
upon  the  market  by  The  Leeds  and  Northrup  Company  of  Phila- 
delphia, Pa.  In  their  highest  grade  apparatus,  constructed  accord- 
ing to  the  above  plan  and  used  with  the  set  of  ratio  coils,  shown  in 
Fig.  612b,  resistances  may  be  compared  in  the  range  of  from  1  to 
10~5  ohm,  and,  with  reduced  precision,  considerably  lower.  The 
precision  of  the  measurement  is  a  maximum  when  the  standard 
and  resistance  under  comparison  are  about  equal.  In  this  case 
0.02  of  1  per  cent  or  better  may  be  expected. 

The  utilization  of  the  Kelvin-double-bridge  principle  is  not  con- 
fined to  the  employment  of  apparatus  (often  of  high  cost)  specially 
constructed  for  the  purpose.  One  standard  of  low  resistance  pro- 
vided with  potential  terminals  must  be  available.  The  rest  of 
the  apparatus  may  be  assembled  from  boxes  of  resistance  coils, 
such  as  are  found  in  most  well-equipped  laboratories.  What  may 
be  accomplished  with  such  facilities  is  best  shown  by  giving  a 
brief  description  of  some  measurements,  made  upon  a  bar  of 
magnesium,  as  an  exercise  for  students. 

In  all  that  has  been  said  above  no  mention  has  been  made  of 
the  importance  of  maintaining  the  temperature  of  the  resistance 
being  measured  at  a  known  value  at  the  time  of  the  measurement. 
When  standard  resistances  made  of  manganin  are  to  be  compared, 
comparatively  rough  measurements  of  temperature  will  suffice; 
but,  in  the  measurement  of  the  resistance  of  a  pure  metal,  the 
temperature  must  be  very  carefully  determined,  hence  the  pre- 
cautions taken  in  this  regard  in  the  sample  measurement  given  in 
par.  614. 

613.  Plan  of  Procedure  for  Making  and  Recording  a  Measure- 
ment. —  It  is  important  when  one  comes  to  plan,  execute,  and 
record  an  electrical  measurement  to  follow  a  systematic  procedure. 
As  the  result  of  the  author's  experience  he  has  found  the  following 
outline  to  embody  the  most  satisfactory  plan  to  follow  in  recording 


ART.  614]      THE   MEASUREMENT  OF  LOW   RESISTANCE  127 

an  electrical  measurement :  (Following  this  plan  is  an  illustration 
of  a  record  of  a  sample  measurement  to  determine  the  resistivity 
of  magnesium.) 

Date  of  experiment  or  measurement  and  names  of  observers. 

Object  of  the  experiment  or  measurement. 

The  precision  sought. 

The  sample  to  be  measured;   described. 

The  method  of  measurement;  described  or  reference  given  to  a 
description. 

Theory  of  method;  explain  or  give  book  references.  Give 
formulae  to  be  used  in  deduction  of  results. 

The  apparatus ;  briefly  described.    Give  instrument  numbers,  etc. 

The  electrical  connections  used;  make  complete  and  clear 
diagrams. 

Procedure.  State  clearly  the  way  in  which  the  observations 
were  made. 

The  data  obtained.  Give  data  clearly  in  tabular  form ;  use  great 
care  not  to  omit  some  one  fact  essential  to  the  deductions. 

Deduction  of  results.  Make  deductions  concise  and  clear  and 
plot  curves  when  several  observations  are  taken  of  more  than  one 
quantity  if  one  quantity  is  a  function  of  another. 

Reliability  and  precision  of  the  measurement.  Very  important 
to  state  clearly  the  probable  precision  attained  and  the  basis 
upon  which  the  probability  of  the  precision  is  judged  to  rest. 

614.  Sample  of  a  Low-resistance  Measurement;  Resistivity 
of  Magnesium.  —  The  object  of  the  measurement  was  to  deter- 
mine the  resistivity  and  temperature  coefficient  of  a  bar  of  mag- 
nesium .metal,  in  the  temperature  range  of  from  20°  C.  to  155°  C. 

The  precision  sought  was  0.2  of  1  per  cent  in  the  final  result. 

The  sample  selected  for  the  measurement  was  a  bar  of  mag- 
nesium which  had  been  accurately  shaped  in  a  milling  machine 
to  give  it  a  rectangular  cross-section.  Its  dimensions  were  deter- 
mined with  a  micrometer  caliper  and  a  comparator. 

Length  of  bar  over  all  =  35.60  cms. 

Length  of  bar  between  potential  points  =  21.883  cms. 

Breadth  of  bar  =  1.412?  cms. 

Thickness  of  bar  =  1.4448  cms. 

Cross-section  of  bar  =  2.041i  cms.2 

Weight  of  entire  bar  =  78.760  grams. 

Density  of  bar  =  1.7232. 


128  MEASURING  ELECTRICAL  RESISTANCE         [ART.  614 

The  potential  points  were  located  by  drilling  two  holes  in  the 
bar  each  about  6.8  cms.  from  an  end  of  the  bar.  No.  24  brass 
wire  pins  were  driven  into  these  holes  and  to  these  the  potential 
leads  were  soldered. 

The  current  terminals  were  fastened  to  each  end  of  the  bar  by 
means  of  small  brass  clamps  having  jaws  like  a  vise.  Heavy 
copper  leads  were  soldered  to  these  brass  clamps.  As  the  jaws 
of  the  clamps  gripped  the  bar  upon  opposite  sides,  the  current 
entered  the  bar  so  that  the  stream  lines  of  current  very  soon 
became  parallel  with  the  length  of  the  bar  and  were  assumed  to  be 
almost  perfectly  so  at  the  potential  points. 

A  chemical  analysis  of  the  sample  was  made  by  Mr.  H.  E. 
Rankin  of  Princeton  University.  He  found  that  the  magnesium 
was  100  per  cent  pure  within  the  limits  of  error  of  his  analysis. 

The  method  of  measurement  was  the  Kelvin-double-bridge 
method.  For  the  theory  of  the  method  see  par.  610.  • 

The  apparatus  used  consisted  of  the  following:  A  manganin 
standard  resistance  (very  accurate)  of  0.00125  ohm  which  would 
carry  200  amperes  without  undue  heating.  For  the  proportional 
arms,  two  100-ohm  coils  for  the  a  and  a  resistances,  kept  fixed, 
and  for  the  b  and  /3  resistances  two  plug-decade  resistance  boxes 
of  10,000  ohms  capacity  and  variable  in  steps  of  1  ohm.  The 
resistance  in  these  boxes  was  varied  to  secure  a  balance.  The 
source  of  current  was  three  small  storage  cells  joined  in  parallel, 
and  a  rheostat  held  the  current  at  between  10  and  20  amperes. 
A  key  was  used  in  the  battery  circuit.  The  galvanometer  was  a 
Leeds  and  Northrup  H  Form  Galvanometer  of  the  suspended-coil 
D'Arsonval  type.  It  was  undamped  upon  open  circuit.  Its 
approximate  constants  were:  Resistance  of  coil  550  ohms.;  com- 
plete period  7  seconds;  megohm  sensibility  1  scale  division  de- 
flection on  a  scale  1000  scale  divisions  from  its  mirror  with  1  volt 
and  290  megohms  in  circuit. 

Since  it  was  necessary  to  regulate  and  measure  accurately  the 
temperature  of  the  bar,  a  tin  trough  was  provided  in  which  the 
bar  could  be  placed  and  kept  underneath  oil.  Paraffin  oil  (in- 
stead of  kerosene  to  avoid  fire  risk)  was  used.  In  the  bottom  of 
the  trough  was  a  solenoid  of  cotton-insulated  german-silver  wire 
thru  which  current  from  the  110-volt  mains  could  be  passed  for 
the  purpose  of  heating  the  oil.  The  temperatures  were  read  with 
two  accurate  mercury  thermometers. 


ART.  614]      THE   MEASUREMENT   OF  LOW  RESISTANCE 


129 


The  connections  employed  are  given  in  the  diagram  below, 
Fig.  614. 

The  procedure  was  first  to  heat  the  oil  to  about  156°  C.  and 
to  vary  the  ratio  coils  b  and  ft  until  a  rough  balance  was  secured. 
The  current  was  then  cut  off  from  the  heating  coil  and  the  oil 
was  vigorously  stirred  with  a  wooden  paddle.  When  the  temper- 
ature was  uniform  thruout  the  tank  an  accurate  balance  of 
the  bridge  was  obtained  by  varying  6  and  ft  together.  As  the 


M0    bar  (in  oil) 


Standard 


=  100 


Variable 


Ke,      H 


Battery 


FIG.  614. 


galvanometer  was  sensitive  to  changes  in  the  resistances  b,  ft  of 
less  than  one  ohm  and  as  these  resistances  could  not  be  varied  in 
smaller  steps  the  simple  plan  was  adopted  of  setting  the  values  of 
b  and  ft  so  that,  for  the  bridge  to  balance  exactly,  the  oil  must 
cool  down  a  degree  or  two.  Then,  at  the  moment  the  galvanom- 
eter showed  no  deflection  when  the  key  was  closed,  a  reading 
was  taken  of  the  temperature  of  the  oil.  This  reading  of  temper- 
ature and  the  setting  of  b,  (b  =  (3),  were  recorded.  The  oil  was 
then  allowed  to  cool  a  few  degrees  further  and  a  new  reading  was 
taken  with  a  new  setting  of  b  and  ft.  Altogether  37  readings  were 
taken  while  the  oil  cooled  from  154.8°  to  21.4°  C.  As  the  temper- 
ature of  the  oil  approaches  room  temperature  the  cooling  proceeds 
very  slowly  and  the  last  few  readings  were  taken  at  widely  sepa- 
rate intervals.  The  process  might  have  been  hastened  by  pouring 
into  the  can  artificially  cooled  oil. 

After  the  run  was  finished  careful  measurements  were  made  of 
the  resistances  a  and  a  including  the  lead  wires  to  redetermine 
their  resistances.  They  were  found  to  be  each  100  ohms  within 
0.02  of  1  per  cent  and  were  called  exactly  100  ohms.  The  follow- 


130 


MEASURING  ELECTRICAL  RESISTANCE         [ART.  614 


ing  table  exhibits  the  data  obtained  together  with  some  obvious 
deductions. 

Observers  A,  B,  C,  Nov.  13,  14,  and  20, 


Temp, 
degs.  C.° 

b  =  /3  ohms 

Resistivity 
ohms  X  10-« 

Conductivity 
mhos  X  10« 

Remarks 

154.8 

1677 

6.9522 

0.14384 

Cooling  started  here, 

147.3 

1710 

6.8174 

0.14657 

3.07P.M. 

139.0 

1740 

6.7004 

0.14924 

135  8 

1760 

6.6243 

0.15096 

132.7 

1780 

6.5499 

0.15268 

126  8 

1810 

6.4413 

0.15525 

3.17P.M. 

119.9 

1840 

6.3362 

0.15782 

118.9 

1850 

6.3020 

0.15868 

115  1 

1870 

6.2346 

0.16040 

109.8 

1900 

6.1361 

0.16247 

106.1 

1920 

6.0723 

0.16468 

102.8 

1940 

6.0097 

0.16640 

99.2 

1960 

5.9483 

,    0.16812 

95.7 

1980 

5.8881 

0.16983 

92.0 

2000 

5.8294 

0.17155 

88.7 

2020 

5.7716 

0.17326 

3.27P.M. 

85.6 

2040 

5.7150 

0.17498 

82.5 

2060 

5.6595 

0.17669 

79.0 

2080 

5.6056 

0.17841 

3.45  P.M. 

76.0 

2100 

5.5518 

0.18013 

73  0 

2120 

5.4994 

0.18184 

70  1 

2140 

5.4480 

0.18355 

67.3 

2160 

5.3851 

0.18569 

64.6 

2180 

5.3480 

0.18699 

61.7 

2200 

5.2994 

0.18870 

4.00P.M. 

59  0 

2220 

5.2518 

0.19041 

55.5 

2250 

5.1817 

0.19299 

52  6 

2270 

5.1359 

0.19470 

4.20P.M. 

49  6 

2290 

5.0911 

0.19642 

47  9 

2310 

5.0471 

0.19814 

45.0 

2330 

5.0037 

0.19985 

43.0 

2350 

4.9611 

0.20157 

4.40P.M. 

40.5 

2370 

4.9192 

0.20328 

38.1 

2390 

4.8781 

0.20500 

S.OOp.M. 

27.7 

2479 

4.7030 

0.21263 

6.00  P.M. 

21.4 

2537 

4.5954 

0.21761 

11  A.M.  (next  day). 

21.9 

2532 

4.6046 

0.21718 

2.30P.M.  (next  day). 

In  column  (1)  are  recorded  the  temperatures  of  the  oil  taken 
each  time  after  the  oil  had  been  vigorously  stirred.  In  column  (2) 
are  given  the  values  of  b  and  (3.  In  column  (3)  are  given  the  re- 
sistances in  micro-ohms  per  centimeter-cube.  In  column  (4)  are 
given  the  reciprocal  values  of  column  (3),  namely,  the  conductivity 
in  mega-mhos.  In  column  (5)  are  made  some  remarks  relating 
especially  to  the  time  of  taking  the  readings. 

Various  deductions  might  be  made  from  the  above  data.     It 


ART.  614]      THE    MEASUREMENT  OF  LOW  RESISTANCE  131 

is  of  special  interest  to  note  that  the  resistivity  increases  very 
nearly  directly  as  the  temperature.  .  We  are  therefore  justified  in 
finding  the  value  of  the  temperature  coefficient  from  the  formula, 

<*  =  fe^b =  a004189 

when  we  use  the  temperatures  t  =  21.4°  and  T  =  154.8°  and  the 
resistivities  corresponding  to  these  temperatures.  Applying  this 
coefficient  we  find: 

Resistivity  of  magnesium  at  20°  C.  =  4.569  X  10~6  ohm. 

The  probable  precision  of  the  above  results  cannot  be  ascer- 
tained from  this  single  determination,  for  unrecognized  systematic 
errors  may  have  entered.  As  the  measurements  were  made,  how- 
ever, with  much  care  and  all  the  resistances  were  known  to  be  ac- 
curate to  better  than  0.05  of  1  per  cent  the  presumption  is  strong 
that  the  accuracy  of  0.2  of  1  per  cent  aimed  for,  has  been  obtained. 
It  would,  however,  be  unscientific  to  claim  this  precision  as  having 
been  certainly  obtained.  To  give  the  determination  a  very  high 
probability  of  precision,  within  a  specified  per  cent,  it  would  be 
necessary  to  repeat  the  measurements  with  the  dimensions  of  the 
specimen  changed  and  with  the  use  of  different  ratio  coils,  standard, 
and  thermometer,  if  any  of  these  were  under  suspicion.  As  a 
matter  of  fact,  this  same  sample  was  measured  about  a  year 
previously  at  which  time  it  had  not  been  milled  down  and  its 
dimensions  were  greater.  A  different  standard,  ratio  coils  and 
thermometer  were  also  employed.  The  value  then  found,  for  the 
resistivity  at  20°  C.,  was  4.564  X  10~6  ohm.' 

This  differs  from  the  above  result  by  about  0.11  of  1  per  cent. 
Thus  it  may  be  concluded  with  a  high  degree  of  probability  that 
the  value  last  obtained  is  correct  within  0.2  of  1  per  cent,  the  pre- 
cision sought. 

To  make  the  exhibition  of  the  results  of  the  measurements 
complete,  curves  giving  the  relations  of  resistivity  and  conduc- 
tivity to  temperature  were  plotted  upon  a  sheet  of  cross-section 
paper  of  large  size. 

In  any  case  where  a  number  of  observations  are  made  of  two 
quantities,  where  one  is  a  function  of  the  other,  the  most  probable 
relations  between  the  quantities  are  best  exhibited  by  means  of 
a  curve.  The  curves  drawn  belonging  to  the  above  measurement 
are  not  given  here  because  of  lack  of  space. 


CHAPTER  VII. 

THE    DETERMINATION    OF    ELECTRICAL 
CONDUCTIVITY. 

700.  Standards  of  Conductivity;  Their  Relation.  Useful 
Formulae.  —  The  determination  of  electrical  conductivity  has 
become  commercially  important  because  the  money  value  of  con- 
ductors, as  conveyers  of  electrical  energy,  is  directly  proportional, 
other  things  being  equal,  to  the  conductivity  of  their  material. 
There  is,  however,  a  certain  confusion  of  ideas  respecting  the 
precise  meaning  of  conductivity,  which  arises  from  the  use  of  three 
different  standards  for  this  property.  It  is  therefore  important 
to  preface  our  description  of  the  methods  of  determining  con- 
ductivity by  a  statement  of  the  definitions  of  the  standards  em- 
ployed and  by  showing  the  mathematical  relations  which  connect 
them. 

Conductance  and  resistance  are  terms  which  apply  to  the  elec- 
trical properties  of  an  electrical  circuit. 

Conductance  =  — : 

resistance 

Conductivity  and  resistivity  are  terms  which  apply  to  specific 
electrical  properties  of  a  conductor. 

Conductivity  =  — .  ,.   .,    • 
resistivity 

If  p  denotes  resistivity,  namely,  the  resistance  between  opposite 
faces  of  a  centimeter-cube  of  the  material,  and  S  denotes  the  cross- 
section  (supposed  uniform)  of  the  conductor,  and  R  is  the  re- 
sistance of  a  length  /,  then 

P-jR.  (1) 

Hence  if  a  is  the  conductivity,  as  above  defined, 

RS'  (2) 

Now  the  resistance  R  cannot  be  considered  constant  as  it  varies 
with  the  temperature.  Hence  if  RQ  is  the  resistance  of  the  con- 

132 


ART.  700]  ELECTRICAL  CONDUCTIVITY  133 

ductor  at  the  temperature  0°  C.  and  6  is  a  coefficient  by  which 
RQ  must  be  multiplied  to  obtain  the  resistance  at  some  other 
temperature  i  of  the  conductor  we  have 


whence,  =  „  =  __.  (3) 


According  to  Eq.  (3)  conductivity  is  a  specific  property  of  the 
material  and  is  a  function  of  the  temperature.  As  resistivity  is 
expressed  in  ohms,  and  as  conductivity,  according  to  Eq.  (3)  is 
the  reciprocal  of  resistivity,  it  is  expressed  in  the  unit  called  the 
mho,  cubic  centimeter.  (Ohm  written  backwards.) 

It  becomes  convenient,  however,  to  treat  of  the  property  of  a 
conductor,  whereby  its  quality  as  a  conveyer  of  electrical  energy 
is  considered,  in  comparison  with  the  property  of  another  con- 
ductor taken  as  a  standard  in  this  respect.  We  shall  then  find, 
that,  in  the  same  way  as  specific  gravity  is  the  relation  of  the 
density  of  a  substance  in  its  actual  state  to  the  density  of  water 
under  specified  conditions,  so  conductivity  may  be  considered  to 
be  the  relation  of  the  conductance  of  one  conductor  in  its  actual 
state  to  that  of  another  conductor  of  like  length  and  cross-section 
or  like  length  and  mass,  which  has  been  selected  as  the  standard. 

This  point  of  view  leads  to  two  other  definitions  of  conductivity, 
according  as  the  sample  selected  for  the  comparison  is  reduced 
to  a  conductor  of  like  length  and  cross-section  or  to  a  conductor 
of  like  length  and  mass,  with  the  standard.  The  advantage  to  be 
gained,  in  defining  conductivity  as  the  ratio  of  two  conductances, 
results  from  the  fact  that  two  conductors,  which  have  the  same 
temperature  coefficients,  may  be  compared  in  respect  to  their  con- 
ductances without  determining  the  temperature  of  either.  It  is 
then  only  necessary  to  be  assured  first,  that  the  temperature  co- 
efficients are  sufficiently  near  alike  for  making  the  comparison 
and  second,  that  when  the  comparison  is  made  both  conductors, 
sample  and  standard,  have  the  same  temperature. 

When  the  conductor  selected  for  a  standard  is  specified  in  a  par- 
ticular way  with  respect  to  resistance  at  a  particular  temperature, 
length  and  cross-section,  or  length  and  mass,  it  is  defined  as  having 
unit  conductivity  without  regard  to  temperature.  If  then  an- 
other conductor  be  reduced  to  like  dimensions,  or  like  length  and 
mass,  and  be  compared  with  this  standard  in  respect  to  con- 


134  MEASURING  ELECTRICAL  RESISTANCE         [ART.  700 

ductance  (the  temperature  of  standard  and  sample  being  merely 
the  same  at  the  time  of  the  comparison)  it  will  be  found  to  have 
a  conductance  regardless  of  temperature  which  is  a  certain  per 
cent  of  that  of  the  standard.  This  is  called  the  per  cent  conduc- 
tivity of  the  sample. 

Unfortunately  two  standards  of  conductivity  are  in  common  use 
and  this  has  led  to  more  or  less  confusion.  These  two  standards 
are  called  Matthiessen's  standards  and  were  recommended  by  the 
Standard  Wiring  Table  Committee,  Jan.  17,  1893.  They  are  thus 
defined : 

Matthiessen's  standard  with  respect  to  diameter  is:  a  copper 
wire  of  circular  and  uniform  cross-section,  1  meter  long  and  1  milli- 
meter in  diameter,  which  has  a  resistance  at  0°  C.  of  0.0203  inter- 
national ohm. 

This  definition  applies  also  to  a  wire  of  uniform,  but  not  circular 

cross-section  when  the  cross-section  is  -  mm2. 

4 

Matthiessen's  standard  with  respect  to  mass  is:  a  copper  wire 
of  uniform  cross-section,  1  meter  long,  which  has  a  mass  of  1  gram 
and  a  resistance  at  0°  C.  of  0.14173  international  ohm. 
'  The  connecting  link  between  these  two  standards  is  the  density 
of  copper.  For  the  two  standards  to  be  equivalent,  that  is,  to  be 
alike  in  specific  resistance  when  at  the  same  temperature,  the 
copper  of  the  meter-gram  standard  must  have  a  particular  density. 
To  show  this,  call  A  the  meter-millimeter  standard,  and  B  the 
meter-gram  standard,  and, 

let    I  —  the  length  of  standard  A} 

S  =  the  cross-section  of  standard  A , 
D  =  the  diameter  of  standard  A, 
pa  =  the  resistivity  of  the  copper  of  standard  A, 
8  =  the  density  of  the  copper  of  standard  B, 
pb  —  the  resistivity  of  the  copper  of  standard  B} 
L  —  the  length  of  standard  5, 
s  =  the  cross-section  of  standard  B, 
m  =  the  mass  of  standard  B, 
ra  be  the  resistance  at  0°  C.  of  standard  A,  and 
rb  be  the  resistance  at  0°  C.  of  standard  B. 
Then  we  have 

Pa   =  —I  (4) 

I 


ART.  700]  ELECTRICAL  CONDUCTIVITY  135 


0  , 

or  pa  =     ^^    >  (5) 

srb  f(.\ 

and  Pb  =  -£••  (o; 

m 

Since  s  =  y-7> 

we  have,  also,  p&  =        -  ,  (7) 


If  the  standards  A  and  J5  are  to  be  alike  in  specific  resistance  we 

must  have  pb  =  pa,  or 

mrb      TT  D2ra  ,  . 

~ 


or  we  must  have 

4  lmrb 


/nN 

Now  in  Eq.  (9)  we  have,  by  the  definitions  of  standards  A  and  B, 
I  =  L  =  100  cms., 
m  =  1, 

r6  =  0.14173  ohm, 
ra  =  0.0203  ohm, 
D  =  0.1  cm., 
TT  =  3.1416, 
whence, 

S  -  _  4  X  100  X  014173  ---  g  ^ 

3.1416  X  O.I2  X  100  X  0.0203 

which  equals  8.89  within  0.0056  of  1  per  cent. 

It  may  be  noted  that  the  meter-gram  standard,  when  in  the 
form  of  a  wire  of  circular  cross-section  of  density  8.89,  is  0.3785  mm. 
in  diameter. 

The  value  of  the  meter-millimeter  standard  when  at  0°  C.,  when 
expressed  in  mhos  is  readily  obtained  from  Eq.  (5).  Thus,  call- 
ing O-Q  its  conductivity  in  mhos,  we  have 

° 


3.1416  XO.PX  0.0203 

Likewise  the  conductivity  oV  of  the  meter-gram  standard,  when 
at  0°  C.  and  of  density  8.89,  is  obtained  from  Eq.  (7)  and  is 

=  0.6273  XIV.  (ID 


pb      mrb      1  X  0.14173 


136  MEASURING  ELECTRICAL  RESISTANCE         [ART.  700 

which  value  equals  the  other,  as  it  should,  when  the  density  is 
taken  as  8.89. 

The  value  of  either  standard  in  mhos  at  other  temperatures 
than  0°  C.  may  be  found  if  we  know  the  temperature  coefficient 
of  the  copper  of  which  the  standard  is  made. 

The  Standard  Wiring  Table  Committee  in  defining  the  Matthies- 
sefn's  standards  did  not  specify  the  coefficient  which  the  copper 
standards  should  have.  But  calling  &  =  1  +  at,  where  a  is  the 
ordinary  temperature  coefficient  of  resistance,  the  value  of  Matthies- 
sen's  standards  at  any  temperature  expressed  in  mhos  becomes 


„  =    -g.  (12) 

We  shall  now  give  some  further  useful  relations  between  the 
general  and  the  specific  properties  of  conductors.  It  is  first 
required  to  find  an  expression  for  the  conductivity  of  any  homo- 
geneous conductor  of  uniform,  circular  cross-section  when  referred 
to  Matthiessen's  meter-millimeter  standard. 

The  ohmic  resistance  rT  of  any  conductor  at  temperature 
T  and  temperature  coefficient  6',  where  this  is  defined  as  the 
ratio  of  the  resistance  rT  at  temperature  T  to  the  resistance  r0  at 
temperature  0°  C.,  is 

rT  =  ~k0.  (13) 

Here  L  is  the  length  and  D  the  diameter  of  the  conductor,  and 
kQ  is  a  constant  which  will  depend  for  its  value  both  upon  the 
nature  of  the  material  and  upon  the  units  chosen  for  the  length 
and  diameter  of  the  conductor.  From  Eq.  (13) 

1        4L0' 


where  CTO  now  expresses  the  conductance  of  the  conductor  at  0°  C. 
Similarly,  we  would  have  for  the  conductance  at  0°  C.,  of  any 
other  conductor  of  circular  cross-section, 


Here  I  is  the  length  and  d  the  diameter  of  the  conductor.     Rt  is 

r> 

the  resistance  of  the  length  I  at  t  degrees  and  6  =  —}  its   tempera- 

-fi/o 

ture  coefficient.     Now,  by  definition,  conductivity,  is  the  ratio  of 
the  conductance  of  one  conductor  to  the  conductance  of  another 


ART.  700]  ELECTRICAL  CONDUCTIVITY  137 

conductor  taken  as  the  standard,  when  the  comparison  is  reduced 
to  a  comparison  between  conductors  of  like  length  and  diameter 
at  the  same  temperature. 

Calling  c  conductivity   (the  unit  of  conductivity  not  as  yet 
being  denned),  we  have 

oV      jB_D^r  (     . 

~  *       Iff  ' 


If  c  is  to  be  expressed  in  terms  of  Matthiessen's  meter-millimeter 
standard,  and  is  called  C8,  then  we  shall  have  O-Q  of  standard  value 
and  c  =  C8,  when, 

0'  =  1,     rT  =  r0  =  0.0203,    and   D  =  1    millimeter   and   L  = 
1  meter.     Thus 


Eq.  (17)  enables  the  conductivity  by  Matthiessen's  meter-milli- 
meter standard  to  be  calculated  from  a  resistance  measurement  of 

n 

a  wire  of  circular  cross-section,  of  temperature  coefficient,  6  =  -—  » 

KQ 

of  diameter  d  millimeters,  of  length  I  meters  and  of  measured 
resistance  at  t°  C.  of  Rt  ohms  for  I  meters. 

The  expression  for  the  conductivity  of  any  conductor,  when 
referred"  to  Matthiessen's  meter-gram  standard,  may  be  derived 
as  follows: 

The  mass  of  a  length  I  of  any  conductor,  having  a  uniform  cross- 
section  S  is  m  =  k'lS,  where  k'  is  a  constant.  The  ohmic  re- 
sistance at  t°  C,  of  any  conductor  is 

k0W 

Rt=  ~S~ 

where  /c0  is  a  constant  which  depends  both  upon  the  units  of  length 
and  cross-section  chosen  and  upon  the  nature  of  the  material, 

r> 

and  9  =  -^  is  the  temperature  coefficient.     Placing  the  value  of 
/io 

S  found  from  the  former  expression  in  the  latter  gives 

VkJ*  .„ 

.       B,  =  -jj-  (18) 

If  we  choose  I  =  1,  m  =  1,  and  t  =  0°  C.,  then  0  =  1,  and  R,  = 
/to  =  k  kot 

p 
and  /tf 


138  MEASURING  ELECTRICAL  RESISTANCE         [ART.  700 


= 

is  the  conductance  at  0°  C.  of  a  conductor  of  length  /,  mass  m 
and  resistance  Rt,  at  t°  C. 

Similarly  for  any  other  conductor 

(20) 


is  the  conductance  at  0°  C.  of  a  conductor  of  length  L,  mass  M 
and  resistance  r/  at  t°  C. 

A  conductivity  being  the  ratio  of  two  conductances,  we  have 
C         W 


,     } 

C'      mRt  LW 

Now  C'  is  a  unit  conductance,  according  to  Matthiessen's  meter- 
gram  standard,  when  L  =  1  meter,  M  =  1  gram,  rt'  =  r0f  = 
0.14173  ohm,  and  t  =  0°  C.  in  which  case  0'  =  1.  Using  these 
values  for  L,  M,  rtf  and  0'  in  Eq.  (21)  we  thus  derive,  as  the  expres- 
sion for  the  conductivity  in  terms  of  the  meter-gram  standard, 

n    ._  I2ero'  (99\ 

'    mRt 

It  should  be  noted  that  in  Eq.  (22)  neither  diameters  nor  densities 
appear.  If  the  resistance  of  m  grams  of  wire  of  length  I  meters 
is  known  at  t°  C.,  then  by  Eq.  (22)  we  may  calculate  its  conduc- 
tivity in  terms  of  Matthiessen's  meter-gram  standard,  provided 

6  =  ~  is  known. 

KO 

To  connect  expressions  (17)  and  (22)  write 

7rd2/5 

m  =  --  > 
4 

where  d  is  the  density  of  the  material  which  is  assumed  to  be  in 
the  form  of  a  conductor  of  circular  and  uniform  cross-section 
having  a  diameter  d,  and  a  length  L 

Putting  this  value  of  m  in  Eq.  (22)  gives 


and  taking  the  ratio  of  Eq.  (23)  to  Eq.  (17)  gives 

(24) 


C. 
Giving  in  Eq.  (24)  r0'  and  r0  their  values  according  to  Matthiessen's 


ART.  700]  ELECTRICAL   CONDUCTIVITY  139 

meter-gram  and  meter-millimeter  standards,  which  are  0.14173 
and  0.0203  respectively,  we  have  the  expression, 


The  relation  given  in  Eq.  (25)  enables  the  conductivity  of  a  metallic 
conductor,  when  found  by  any  method  and  expressed  in  terms 
of  one  of  Matthiessen's  standards,  to  be  expressed  in  terms  of 
the  other,  provided  the  density  of  the  material  is  known.  If  the 
material  of  the  conductor  has  a  density  the  same  as  the  stand- 
ard, namely  8.89,  then  its  conductivity  will  be  expressed  by  the 
same  number  whether  it  be  referred  to  the  meter-millimeter  or  to 
the  meter-gram  standard.  This  will  be  the  case  for  the  accuracy 
required  in  engineering,  when  expressing  the  conductivities  of 
samples  of  copper.  This  being  so,  a  statement  of  the  particular 
standard  to  which  the  conductivity  of  the  copper  is  referred  is 
often  omitted.  When  the  material,  however,  is  iron  or  aluminum 
having  a  density  different  from  that  of  copper,  it  is  essential  to 
state  to  which  standard  its  conductivity  is  referred.  The  author 
believes  that  there  would  be  a  gain  in  simplicity  and  an  avoidance 
of  considerable  confusion  if  one  standard,  the  meter-gram  stand- 
ard, were  exclusively  employed  in  all  engineering  practice. 

We  shall  illustrate  by  numerical  examples  the  uses  of  Eqs.  (17), 
(22),  and  (25)  and  then  describe  the  methods  in  use  for  deter- 
mining conductivities. 

Example  1.  —  Conductivity   of   Magnesium. 

By  Eq.  (17)  conductivity  by  the  meter-millimeter  standard  is 
r  _W  0.0203 


Referring  to  the  data  in  par.  614  for  magnesium  we  find  Rt  = 
#20  =  4.569  X  10"6  ohm  for  the  resistance  between  opposite  faces 
of  a  centimeter  cube.  Now  0=l+c*20  =  l+  0.004189  X 
20  =  1.0838.  If  A  is  the  cross-section  of  the  sample,  which  in 
this  case  is  1  sq.  cm.  or  100  sq.  mm.,  we  have 

inn      v&       u  M      40° 

A  =  100  =  --  >     hence     d2  =  --  > 

4  7T 

also  I  =  0.01  meter.  Using  these  values  in  the  general  formula 
above  we  obtain 

0.01  X  1.0838  X  0.0203  X  3.1416 


QCTxT569  X  10- 


=  0.3782  conductivity, 


140  MEASURING  ELECTRICAL  RESISTANCE         [ART.  701 

or,  as  this  quantity  is  usually  stated,  37.82  per  cent  conductivity 
according  to  Matthiessen's  meter-millimeter  standard.  By 
Eq.  (25),  the  conductivity  by  Matthiessen's  meter-gram  standard 
is 

8.89 


Since  the  density  found  for  magnesium  is  d  =  1.7232  we  find 

8  89 

Cw  =     ~OQO  X  0.3782  =  1.951  conductivity, 
1. 


or  195.1  per  cent  conductivity  according  to  Matthiessen's  meter- 
gram  standard.  Thus  for  corresponding  cross-sections  magnesium 
is  0.3782  times  as  good  a  conductor  as  copper  and  for  correspond- 
ing masses  it  is  1.951  times  as  good  a  conductor. 

Example  2.  —  Conductivity  of  Aluminum. 

See,  for  the  example,  Appendix  II,  5,  Example  2. 

Example  3. 

See,  for  the  example,  Appendix  II,  5,  Example  3. 

701.  The  Measurement  of   Conductivity.  —  The  commercial 
measurement  of  conductivity,  in  terms  of  one  of  Matthiessen's 
Standards,  involves  the  comparison  of  the  resistance  of  a  low- 
resistance  wire  having  a  large  temperature  coefficient  with  the 
resistance  of  another  low-resistance  wire  having  approximately 
the  same  temperature  coefficient.     If  the  standard  and  sample 
wires  are  maintained  at  the  same  temperature  and  both  have 
nearly  the  same  temperature  coefficient,  the  comparison  can  be 
made  quite  accurately  without  a  knowledge  of  the  actual  tem- 
perature.    The  method  and  apparatus  best  adapted  commercially 
to  a  rapid  and  accurate  determination  of  conductivity  were  both 
invented  by  Mr.  Wm.  Hoopes  and  the  apparatus  is  generally  known 
as  the  "  Hoopes  Bridge." 

702.  The  Hoopes  Bridge  for   Conductivity  Determinations; 
Described.  —  The  essential  feature  of  the  method  consists  in  an 
adaptation  of  the  Kelvin-double-bridge  principle  to  a  bridge  fitted 
with  a  standard  conductor  and  special  scales  from  which  the  per 
cent  conductivity  of  the  sample,  inserted  in  the  bridge,  is  read 
directly,   without  calculation.     This  method  may  be  explained 
as  follows: 

From  the  Kelvin-double-bridge  arrangement,  exhibited  in  Fig. 

702a,  it  is  evident  that  when  the  ratio  -  =  —  ,  we  have  the  resistance 

P      Pi 


=  r-p, 
p 


ART.  702]  ELECTRICAL  CONDUCTIVITY  141 

X  between  the  potential  points  c  and  d  given  by  the  relation, 

CD 

where  P  is  the  resistance  included  between  the  potential  points  a 

V 

and  6.     In  practice  -  is  made  equal  to  unity.     Both  r  and  p  are 

chosen  equal  to  300  ohms,  in  order  that  the  small  resistances  at 
the  knife-edge  contacts  used  for  potential  points,  shall  be  very 


eft 


J100| 


I 

FIG.  702a. 

small  in  comparison  with  the  resistance  of  the  ratio  coils  and  hence 
negligible.  A  galvanometer  of  about  300  ohms  resistance  and 
200  megohms  sensibility  is  used  with  the  bridge.  When  the  gal- 
vanometer indicates  that  the  bridge  is  balanced 

X  =  P.  (2) 

Referring  to  Fig.  702a,  suppose  that  X  is  a  rod  of  copper  having 
a  cross-section  S.  Suppose  the  scale  /  to  have  100  equal  divisions 
of  arbitrary  length  between  the  points  0  and  100.  The  resistance 
and  its  reciprocal,  the  conductance,  of  the  rod  X  will  depend  upon 
the  purity  and  the  physical  condition  in  regard  to  hardness, 
temperature,  etc.,  of  the  copper  of  which  it  is  made. 

Now  suppose  that  P  is  also  a  rod  of  copper  having  a  uniform, 
but  not  necessarily  known,  cross-section.  Suppose  the  points  a 
and  b  are  separated  until  they  include  a  length  of  the  rod  P  which 
will  have  the  same  resistance  as  the  resistance  of  the  rod  X  of 
cross-section  S,  of  100  per  cent  conductivity,  and  a  length  equal 
to  100  divisions  of  scale  7.  This  being  done,  the  bridge  will  be 
balanced.  Now,  if  no  change  is  made  in  P  or  in  the  position  of 
the  contacts  a  and  6,  and  we  substitute  for  X  another  rod  X'  of 
the  same  cross-section  but  of  less  conductivity,  it  will  then  be 


142  MEASURING  ELECTRICAL  RESISTANCE         [ART.  702 

necessary,  in  order  to  balance  the  bridge,  to  move  the  contact  d 
towards  the  zero  mark  of  scale  /.  Suppose  the  second  rod  X' 
has  twice  the  resistance  of  X,  or  half  the  conductivity,  then  d  will 
have  to  be  moved  for  obtaining  a  balance  half  way  down  the 
scale  7,  that  is,  to  the  50-di vision  mark.  Hence,  in  this  case  the 
50-division  mark  indicates  that  rod  Xf  has  50  per  cent  conductivity 
according  to  a  standard  based  on  cross-section. 

Again,  suppose  we  substitute  for  the  rod  X  a  rod  Xi,  which  has 
100  per  cent  conductivity,  but  twice  the  cross-section  of  rod  X. 
In  order  to  obtain  a  balance  it  will  be  necessary  now  to  make  the 
distance  between  a  and  b  one-half  what  it  was  before.  Then  set 
b  at  the  middle  point  of  scale  H.  If  now  we  substitute  for  X\  a 
rod  Xi  having  the  same  dimensions,  but  of  lower  conductivity 
than  Xij  a  balance  will  be  obtained  by  moving  d  toward  the  zero 
mark  of  scale  /,  and  the  reading  of  the  scale  will,  as  in  the  former 
case,  give  the  per  cent  conductivity  of  Xi,  which  has  twice  the 
cross-section  of  the  original  rod  X.  Thus  we  may  find  a  series  of 
positions  on  the  scale  H  corresponding  to  rods  to  be  tested  of 
various  diameters  or  cross-sections. 

Having  a  rod  to  test  of  a  particular  diameter,  the  slider  b  is 
set  to  a  division  on  scale  H  corresponding  to  that  particular  diam- 
eter, then  the  reading  on  scale  /  gives  directly,  a  balance  being 
obtained,  the  per  cent  conductivity  of  the  sample  being  tested. 
The  Kelvin  double  bridge,  fitted  with  the  Hoopes'  scales,  as  above 
described,  gives  the  conductivity  in  terms  of  the  meter-millimeter 
standard.  As  this  bridge  is  actually  constructed  by  its  makers 
at  the  present  time,  the  standard  scale  H,  Fig.  702a,  is  laid  off  in 
gram  weights  instead  of  diameters,  and  the  bridge  readings  are 
then  given  on  the  scale  7  in  per  cent  conductivities  based  upon 
Matthiessen's  meter-gram  standard.  The  bridge  gives  by  a  direct 
reading  the  value  designated  in  Eq.  (22),  par.  700,  by  the  sym- 
bol Cw,  multiplied  by  100. 

In  Fig.  702b  are  given  views  of  Hoopes'  conductivity  bridge  out 
of  and  in  its  inclosing  metal  case.  In  practice  it  is  always  used  in 
its  case,  which  is  made  of  metal  to  insure  equality  of  temperature 
thruout  its  interior.  The  bridge  measures  the  conductivity  of 
wires  from  No.  0000  B.  &  S.,  to  No.  18  B.  &  S.  gauge. 

It  may  be  supplied  with  different  standards  to  take  care  of 
wires  of  different  sizes  and  various  kinds.  One  standard  covers 
a  range  of  three  wire-sizes,  B.  &  S.  gauge.  The  standard  wires 


ART.  703] 


ELECTRICAL  CONDUCTIVITY 


143 


are  made  of  the  same  material  as  the  samples  to  be  measured  to 
insure  an  equality  between  the  temperature  coefficients  of  standard 
and  samples. 


FIG.  702b. 


703.   The  Hoopes  Bridge;  Operations  Required  for  Using. — 

"  The  actual  operation  of  making  a  measurement  is  so  simple  that 
it  scarcely  requires  explanation.  The  sample  to  be  tested  is 
placed  in  a  cutting-off  machine  and  cut  to  the  standard  length. 
It  is  then  weighed  to  within  an  accuracy  of  0.05  of  1  per  cent, 
and  inserted  in  the  bridge  under  the  clamps  and  drawn  out 
straight,  but  not  stretched,  by  a  gripping  tool  provided  for  the 
purpose. 

The  contact  on  the  standard  is  placed  at  a  setting  corresponding 
to  the  weight  in  grams  of  the  sample.  The  box  is  then  closed 
and  a  few  minutes  allowed  to  elapse  to  give  the  sample  time  to 
acquire  the  temperature  of  the  standard.  The  sliding  contact  is 
then  moved  back  and  forth,  until  a  balance  is  obtained.  The 
metal  lid  covering  the  glass  in  front  of  the  box  is  then  raised  and 
the  scale  reading  taken.  This  reading  is  the  required  per  cent 
conductivity  of  the  sample  tested.  If  many  samples  are  to  be 


144  MEASURING  ELECTRICAL  RESISTANCE         [ART.  704 

tested  they  may  all  be  cut  off  at  one  time,  weighed,  tagged  and 
left  in  the  rack  in  the  box  while  measurements  are  being  made. 
It  will  then  require  but  two  or  three  minutes  for  the  particular 
sample  under  test  to  acquire  the  temperature  of  the  standard. 
Ten  samples  an  hour  can  easily  be  measured  in  this  way.  A  larger 
number  of  samples  may  be  measured  per  hour  after  the  operator 
has  gotten  the  measurement  down  to  a  system.  As  high  as 
150  samples  per  day  may  be  tested  with  this  bridge." 

704.  Precautions  to  Observe  in  Using  Hoopes  Bridge.  — 
"The  Hoopes  bridge  gives  accurate  results  only  when  the  temper- 
atures of  the  standard  wire  and  the  sample  are  the  same.  A 
difference  of  1°  C.  will  affect  the  results  about  0.4  of  1  per  cent. 
A  difference  of  temperature  may  be  caused : 

(a)  By  taking  a  reading  too  soon  after  placing  a  sample  in  the 
box,  the  sample  not  having  had  time  to  acquire  the  temperature 
of  the  standard. 

(b)  By  using  too  large  a  current.     This  will  cause  an  unequal 
heating  of  the  sample  and  the  standard  when  the  two  are  not  of 
the  same  dimensions. 

(c)  By  changing  standards.     When  a  new  sample  wire  is  placed 
in  the  box  it  acquires  very  rapidly  the  temperature  of  its  sur- 
roundings.    When,  however,  a  new  standard  bar  is  placed  in  the 
box  it  takes  a  very  long  time  for  the  mass  of  rubber  upon  which 
this  bar  is  mounted  to  take  the  same  temperature  as  the  rubber 
bar  on  which  the  sample  is  placed.     As  the  standard  wire  rests  upon 
the  rubber  of  the  standard  bar  it  will  follow  it  in  its  temperature 
changes.     Hence  when  checking  up  two  standards  by  comparing 
one  with  the  other,  it  is  important  to  make  sure  that  the  standard 
bar  and  the  sample  bar  come  to  the  same  temperature.     This  may 
require  that  a  standard  bar  newly  put  in  the  box  be  left  in  position 
a  long  time  before  a  reading  is  taken. 

As  small  a  current  should  always  be  used  as  will  give  the  neces- 
sary sensibility. 

For  No.  15  standard  wire  use  about  4  amperes. 

For  No.  12  standard  wire  use  about  4  to  5  amperes. 

For  No.  9  standard  wire  use  about  8  to  10  amperes. 

For  No.  6  standard  wire  use  about  11  to  12  amperes. 

For  No.  3  standard  wire  use  about  15  amperes. 

For  No.  0  standard  wire  use  about  18  to  20  amperes. 

For  No.  000  standard  wire  use  about  50  to  100  amperes. 


ART.  706]  ELECTRICAL  CONDUCTIVITY  145 

The  standards  are  checked  between  themselves  by  the  makers 
and  are  presumably  correct.  If  the  bridge  fails  to  give  accurate 
results  the  failure  probably  should  be  laid  not  to  the  standards 
but  should  be  sought  in 

(a)  unequal  temperatures, 

(b)  too  much  current, 

(c)  bad  contacts, 

(d)  inaccurate  weighing, 

(e)  wrong  setting  on  the  standard  scale,  or 

(f)  wrong  cutting  off  of  a  sample." 

More  detailed  descriptions  of  this  bridge  and  its  accessories 
have  been  given  by  the  makers,  The  Leeds  and  Northrup  Com- 
pany of  Philadelphia,  Pa.,  to  whom  the  above  description  is 
due. 

705.  Other  Methods  of  Measuring  Conductivity.  —  As  the 
Hoopes'  conductivity  bridge,  described  above,  is  intended  for  use 
where  a  large  number  of  samples  are  to  be  measured,  and  as  this 
rather  costly  piece  of  apparatus  is  not  always  to  be  had  other 
methods  are  often  employed.  The  method  described  in  par.  614 
under  the  heading,  "  Resistivity  of  Magnesium,"  yields  data  from 
which  the  conductivity  is  readily  calculated  by  the  Eqs.  (17)  and 
(22),  par.  700.  As  Eq.  (17)  involves  Rtj  a  resistance  at  a  tem- 

r> 

perature  t,  and  9  =  ~  ,  a  coefficient  which  must  be  obtained  by 
/to 

measurement  when  not  known,  any  method  which  will  determine 
these  quantities  with  precision  can  be  used  for  determining  con- 
ductivity. If  the  conductivity  is  obtained  in  terms  of  Matthies- 
sen's  meter-millimeter  standard,  then  by  the  relation 

o  on 
CW  =  5HC.  (Eq.  25,  §700), 


the  conductivity  in  terms  of  Matthiessen's  meter-gram  standard 
may  be  calculated  provided  the  density  5  of  the  material  is  known 
or  determined.  However,  as  conductivity  determinations  are  of 
great  importance  commercially,  instrument  makers  have  devised 
special  apparatus  other  than  the  Hoopes  bridge  for  this  purpose, 
which,  tho  no  new  principles  are  involved,  are  adapted  to  yield 
good  results  with  speed  and  precision. 

706.   Equipment     for     Conductivity     Determination;      The 
Standard  Resistance  Variable.  —  The  following  description   of 


146 


MEASURING  ELECTRICAL  RESISTANCE         [ART.  706 


an  equipment  is  taken  in  part  from  literature  printed  by  The 
Leeds  and  Northrup  Company  as  it  contains  certain  useful  infor- 
mation of  a  practical  nature. 

The  equipment  recommended  includes  a  Kelvin  double  bridge  of 
the  type  described  in  par.  612,  Fig.  612b.  In  the  apparatus, 
as  made  by  The  Leeds  and  Northrup  Company,  the  ratio  coils 
giving  ratios  0.1,  1  and  10  are  permanently  mounted  on  a  base 
board  with  a  variable  low-resistance  standard.  The  low-resist- 
ance standard  itself  has  a  total  resistance  of  0.1  ohm  which  may 
be  varied  between  potential  points  by  infinitesimal  steps.  Set- 
tings of  one  potential  point  in  steps  of  0.01  ohm  are  made  by  means 
of  a  plug  inserted  between  blocks  and  a  metal  bar,  and  settings 
of  the  other  potential  point  are  made  by  a  knife-edge  contact 
which  slides  over  a  rod  of  manganin.  This  has  a  resistance  of 
0.01  ohm  between  the  0  and  the  100  mark  of  a  scale  which  lies 
alongside  the  rod. 


UKD 


FIG.  706. 


The  entire  equipment  when  fitted  up  for  speed  and  convenience 
consists  of  the  following: 

The  sliding-contact  Kelvin  double  bridge,  just  described. 

A  galvanometer  which  has  a  sensibility  of  from  100  to  200 
megohms,  and  a  complete  period  of  from  3  to  5  seconds  and  a 
resistance  of  about  200  to  500  ohms. 

Means  for  cutting  off  the  sample  wires  or  rods  to  a  standard 
length.  (A  special  cutting-off  machine  is  sold  for  this  pur- 
pose.) 

Suitable  scales  for  weighing  the  samples. 

Special  clamps  for  holding  the  samples,  and  which  at  the  same 


ART.  707]  ELECTRICAL  CONDUCTIVITY  147 

time  serve  to  lead  the  current  in  and  out  of  the  sample  and  to 
connect  knife-edge  potential  points  to  the  sample. 

A  rheostat  switch  by  which  the  main  circuit  may  be  entirely 
opened  or  closed  thru  a  resistance  which  can  be  diminished 
gradually. 

Two  cells  of  a  storage  battery  of  from  20  to  30  amperes-capacity 
each. 

An  accurate  mercury  thermometer  reading  from  0°  C.  to  60°  C. 
and  graduated  in  tenths  of  a  degree. 

A  suitable  tank  of  metal  to  hold  kerosene  oil  and  long  enough  to 
contain  the  sample. 

The  above  equipment  should  be  assembled  as  shown  in  Fig.  706 
(the  tank  not  being  shown  in  the  figure) . 

707.  Method  of  Using  Variable  Resistance  Standard  for  Con- 
ductivity Determinations.  —  To  make  a  conductivity  determina- 
tion one  should  proceed  as  follows:  The  sample  wire  is  cut  off 
to  some  definite  length,  a  length  of  38  inches  being  recommended. 
The  wire  is  first  weighed  to  within  an  accuracy  of  0.05  of  1  per  cent. 
It  is  then  fastened  in  the  two  clamps  so  as  to  bring  the  two  poten- 
tial points  a  known  distance  apart.  If  the  two  clamps  are  fas- 
tened rigidly  to  a  marble  or  wood  base  the  distance  between  the 
potential  points  will  always  be  the  same  and  need  only  be  meas- 
ured once.  With  a  wire  38  inches  long  it  may  be  taken  28  inches. 
For  the  best  precision  the  wire  and  clamps  should  be  placed  under 
kerosene  oil  in  the  tank.  The  temperature  of  the  oil  is  now  care- 
fully read,  while  it  is  being  stirred,  and  at  the  same  time  the  re- 
sistance of  the  sample  between  its  potential  points  is  determined. 
This  last  operation  is  effected  by  first  adjusting  the  galvanometer 
reading  to  the  zero  of  the  scale  with  the  main  circuit  open.  This 
circuit  is  then  closed  by  means  of  the  rheostat  key  thru  a  high 
resistance  and  the  bridge  is  roughly  balanced  by  adjusting  the  one 
potential  point  of  the  bridge  to  within  0.01  ohm  with  the  plug, 
and  the  other  potential  point  with  the  sliding  contact.  The 
rheostat  key  is  then  moved  so  as  to  cut  out  more  resistance, 
and  a  still  closer  adjustment  for  a  balance  of  the  bridge  is  made 
with  the  sliding  contact.  Finally  the  current  thru  the  bridge  is 
still  further  increased  and  an  exact  adjustment  of  the  bridge  is 
made.  The  current,  however,  should  never  be  made  so  large  as 
to  perceptibly  heat  the  wire.  If  the  current  is  not  heating  the 
wire  perceptibly  the  bridge  will  balance  with  the  sliding  contact 


14$  MEASURING  ELECTRICAL  RESISTANCE        [ART.  708 

at  the  same  point  when  the  current  employed  is  reduced  to  half 
the  value  used  in  obtaining  the  setting.  The  resistance  of  the 
sample  at  the  temperature  t°  C.  is  now 

Rt  =  bridge  setting  X  ratio  plugged. 

Suppose  the  plug  contact  is  set  at  0.07  ohm  and  the  sliding  contact 
at  0.0058,  then  the  bridge  reading  is  0.0758  ohm,  and  if  the  ratio 
plugged  is  0.1  the  resistance  is  Rt  =  0.00758  ohm. 

If  the  sample  is  copper  or  aluminum  and  its  temperature  co- 
efficient is  known,  its  conductivity  may  be  calculated  from  the 
data  obtained  from  the  above  observations.  If  its  temperature 
coefficient  is  not  known  it  must  be  determined  by  again  measur- 
ing the  resistance  of  the  wire  at  some  other  temperature.  The 
temperature  may  be  changed  by  refilling  the  tank  with  oil  which 
is  colder  or  warmer  than  the  oil  used  in  the  first  measurement. 
For  precision  the  oil  in  the  second  case  should  differ  at  least  15°  C. 
from  the  oil  in  the  first  case.  Let  Rt  =  RQ  (1  +  at)  be  the  resist- 
ance when  the  oil  is  at  temperature  t,  and  let  Rt  =  RQ  (1  -f-  ati) 
be  its  resistance  when  at  temperature  h,  then  eliminating  RQ  from 
the  two  relations  above,  we  have 


a  = 
Then  as 


(see  §  700),  this  quantity  is  determined. 

708.  Method  of  Calculating  Conductivity  from  Resistance 
Data.  —  We  can  now  calculate  the  conductivity  in  terms  of  the 
meter-gram  standard  by  Eq.  (22),  par.  700.  This  equation  may 
be  used  as  it  stands,  but  it  may  be  simplified  for  purposes  of  cal- 
culation when  the  samples  used  are  copper  and  these  are  always 
cut  off  to  a  given  length  and  the  potential  points  are  always  set 
the  same  distance  apart. 

It  has  recently  been  shown  by  researches  made  at  the  Bureau  of 
Standards  at  Washington  that  the  coefficient  0  for  copper  increases 
in  direct  proportion  to  the  conductivity  in  the  range  from  0°  to 
100°  C.  For  the  commercial  determination  of  the  conductivity  of 
copper,  however,  we  may  assume  that  this  coefficient  has  at  20°  C. 


ART.  708]  ELECTRICAL  CONDUCTIVITY  149 

the  value  1.0797.*     Putting  in  Eq.  (22),  par.  700,  this  value  of  0 
and  the  value  0.14173  for  rQ'  we  have 

_F  0.14173X1.07968 


Here  R2o  =  the  measured  resistance  in  ohms  of  the  sample  between 
the  potential  points  which  distance  is  to  be  taken  in  meters.  If 
/  is  this  distance,  which  has  been  chosen  28  inches,  we  have  I  =  28 
X  0.0254  =  0.7112  meter.  If  m  is  the  weight  in  grams  of  the 
38  inches  of  sample  cut  off,  then  §|  m  is  the  weight  in  grams  of 
the  length  I  between  the  potential  points.  Hence, 


2 


=  072   X  0.14173  X  1.0797  X  38  =  0.10504 
2SmR20  mR2Q  ' 

Instead  of  using  Eq.  (2)  as  it  stands  we  may  simplify  the  calcula- 
tion by  the  use  of  logarithms. 

Call  K  =  0.10504, 

then, 

log  Cw  =  \ogK-  (log  m  +  log  #20).  (3) 

For  example:  suppose  the  sample  is  a  No.  10  B.  &  S.  gauge 
copper  wire  and  we  find  m  =  45.163  grams,  and  that  the  resist- 
ance at  22°  C.  is  0.00238  ohm.  We  must  first  calculate  the 
resistance  for  20°  C.  This  can  be  done  by  applying  the  known 
temperature  coefficient  for  copper,  which  may  be  taken  0.003984, 
or  we  can  obtain  the  required  value  with  sufficient  exactness  by 
subtracting  from  the  value  of  the  resistance  at  22°  C.  2  X  0.004  = 
0.008  of  this  value.  Thus  R20  =  0.00238  -  0.00238  X  0.008  = 
0.002361  ohm.  (The  more  exact  value  is  0.002363  ohm.) 
We  then  find  by  Eq.  (3) 

Log  Cw  =  log  0.10504  -  (log  45.163  +  log  0.002363), 
or 

LogCw  =  9.0213547  -  10  -  (1.6547828  +  7.3734637  -10), 

or  Cw  =  0.9842  or  Cw  =  98.42  per  cent  conductivity  according 
to  Matthiessen's  meter-gram  standard. 

Tr 

In  the  equation  Cw  =  —  =5  the  constant  K  will  assume  different 
mR 

values  when  R  is  taken  at  different  temperatures.  When  R  is 
taken  at  20°  C.,  the  value  of  K  is  that  in  Eq.  (2).  Where  many 
measurements  are  to  be  made  a  table  should  be  constructed  which 

*  See  later  value  given  in  circular  No.  31,  "Copper  Wire  Tables,"  issued 
April  1,  1912  by  the  Bureau  of  Standards. 


150 


MEASURING  ELECTRICAL  RESISTANCE         [ART.  709 


will  give  the  values  which  K  assumes  when  R  is  taken  at  different 
temperatures.  It  will  then  not  be  necessary  always  to  reduce 
the  resistance  to  its  value  at  20°  C. 

709.  Conductivity  Determinations  with  Fixed  Resistance 
Standard  and  Variable  Ratios.  —  For  measuring  the  resistance 
of  the  sample  a  fixed  standard  low  resistance,  used  in  connection 
with  variable  ratio  coils  (see  §  612)  may  be  employed.  The  pro- 
cedure would  then  be  exactly  the  same  as  in  the  case  described 
above,  except  that  the  bridge  would  be  balanced  by  varying  ratio 
coils  instead  of  a  low  resistance.  The  fixed  standard  resistance 
should  be,  preferably,  of  approximately  the  same  resistance  as 
the  sample. 


«" 7 


M 


FIG.  709. 


A  set  of  variable  ratio  coils,  as  made  by  Otto  Wolff  of  Berlin 
and  described  in  par.  612,  may  be  used  for  conductivity  determina- 
tions very  advantageously  as  follows:  The  method  of  using  the 
ratio  coils  is  planned  to  avoid  the  necessity  of  measuring  the  tem- 
perature of  the  sample  by  employing  a  standard  made  of  the  same 
material  as  the  sample  when  making  the  comparison.  The  method 
is  then  similar  in  principle  to  the  Hoopes  bridge  method  described 
in  par.  702. 

A  standard  rod  of  the  same  material  as  the  sample  to  be  meas- 
ured is  prepared.  This  is  provided  with  .two  potential  terminals 
(set  at  or  soldered)  a  fixed  distance  apart.  This  standard  is  con- 
nected with  the  variable  ratio  coils  and  sample  to  form  a  Kelvin 
bridge  as  shown  in  Fig.  709. 

The  sample  and  standard  should  be  laid  close  to  one  another, 
and  perhaps  be  covered  so  they  will  assume  the  same  tempera- 
ture. The  connection  D  should  be  given  as  low  a  resistance  as 
practicable.  The  rheostat  key  Rh  may  be  used  with  additional 


ART.  709]  ELECTRICAL  CONDUCTIVITY  151 

convenience.  When  a  balance  of  the  bridge  has  been  obtained, 
the  conductivity  by  the  meter-gram  standard  is  readily  calculated 
from  the  following  data  and  relations. 

Let  M  =  weight  of  the  standard  between  its  potential  points. 

Let  L   =  distance  between  potential  points  of  standard. 

Let  Sw  =  conductivity  of  the  standard. 

Also  let  m,  I,  and  Cw  be  the  corresponding  quantities  for  the 
sample  to  be  measured. 

Then  by  Eq.  (22)  par.  700,  we  have 

<?*,  =  —>  (1) 

and 


„  .„. 

*•    SB?' 

If  standard  and  sample  are  of  the  same  material  and  are  at  the 
same  temperature  we  have 

Cw  =  Ml2  Rtf 

Sw      mL2  Rt' 
Now,  by  the  equation  of  the  Kelvin  bridge, 


Rt      a 
and  we  have, 


It  will  often  be  possible  to  choose  I  of  such  a  length  that  the 

product  Ml2  =  mL2,  in  which  case  Cw  =  -  Sw.     Or  at  least  one 

a 

can  make  I  =  L.  in  which  case  Cw  =  -  —  Sw. 

a  m 

This  method  is  to  be  recommended  when  one  possesses  a  set  of 
Otto  Wolff  variable  ratio  coils.  If  the  standard  is  accurate  it  will 
give  very  accurate  results  and  the  calculations  are  very  simple. 
In  conclusion  it  should  be  remarked,  that  any  method  which  will 
measure  with  accuracy  the  resistance  of  a  short  rod  or  wire  at 
two  temperatures  will  give  the  necessary  data  from  which  the 
conductivity  may  be  deduced.  When  the  density  of  the  material 
is  known,  the  conductivity  may  be  expressed;  in  mho,  cubic  centi- 
meter, units,  or  as  a  per  cent  conductivity  of  either  Matthiessen's 
meter-millimeter  or  meter-gram  standard.  For  scientific  purposes 
the  first  is  to  be  preferred  and  for  commercial  purposes  the  last. 


CHAPTER  VIII. 
THE  MEASUREMENT  OF  HIGH  RESISTANCE. 

800.  High  Resistance  Specified  and  Described.  —  If  a  Wheat- 
stone  bridge  has  ratio  arms  which  will  give  a  ratio  of  10,000  to  1 
and  a  rheostat  which  reads  to  10,000  ohms,  then  a  resistance  of 
108  ohms  or  100  megohms  may  be  measured,  theoretically.  But 
the  insulation  of  the  bridge  would  need  to  be  very  high  and  the 
ratio  coils  very  accurately  adjusted  to  assure  even  moderately 
precise  results.  When  a  resistance,  therefore,  exceeds  10  megohms 
it  can  be  more  conveniently  and  more  accurately  measured  by 
some  one  of  the  methods  which  have  been  devised  specially  for 
the  purpose.  We  shall  treat  all  resistances  as  "  high  resistances  " 
which  exceed  10  megohms  and  give  the  methods  best  adapted  for 
their  measurement. 

Practically  all  metallic  resistances  are  excluded  from  the  class 
"  high  resistances,"  because  it  is  rare  to  find  a  single  metallic 
resistance  unit  which  exceeds  a  megohm. 

The  methods  to  be  described  are  adapted  to  the  measurement 
of  high  resistances  which  may  be  considered  under  two  general 
classes :  First,  resistances  which  are  not  associated  with  an  appreci- 
able electrostatic  capacity  and  which  obey  exactly  or  approximately 
Ohm's  law.  Such  high  resistances  are :  the  insulation  resistance  of 
electrical  apparatus,  the  resistivity  of  insulating  materials,  the 
insulation  over  a  surface  of  insulating  material,  or  any  high 
resistance  where  the  effect  of  capacity  need  not  be  taken  into 
account  in  making  the  measurement.  Under  this  class,  also,  may 
be  considered  the  special  methods  which  are  required  for  deter- 
mining the  insulation  resistance  of  a  wiring  system  while  the  power 
is  on. 

Second,  high  resistances  which  are  associated  with  more  or  less 
electrostatic  capacity,  the  presence  of  which  causes  the  resistance 
to  act  as  if  it  did  not  obey  Ohm's  law,  at  least  in  the  first  few 
moments  after  the  current  is  made  or  broken.  This  class  includes 
the  resistance  of  condensers  and  the  insulation  resistance  of  long 
cables.  For  the  determination  of  such  resistances  special  methods 

152 


ART.  801J      THE  MEASUREMENT   OF  HIGH  RESISTANCE         153 


of  procedure  and  specifications  must  be  adopted  to  obtain  con- 
cordant results  which  shall  have  a  precise  significance. 

While  the  accurate  measurement  of  ordinary  resistances  is 
generally  made  by  some  null  or  balance  method,  high  resistances, 
on  the  other  hand,  are  generally  measured  by  some  type  of  de- 
flection method.  Two  general  methods,  of  which  there  are  many 
modifications  in  detail,  are  in  use;  one  in  which  the  high  resistance 
is  measured  in  terms  of  the  deflections  of  a  voltmeter,  galvanom- 
eter, electrometer  or  like  instrument,  and  one  in  which  the  high 
resistance  is  determined  by  the  time  required  for  the  charge  given 
to  a  condenser  to  leak  partly  away  thru  the  resistance  being 
measured.  A  high  resistance  of  a  few  hundred  megohms  may 
also  be  measured  by  a  balance  method  employing  the  principle  of 
the  Wheatstone  bridge,  when  one  has  a  sensitive  galvanometer 
and  a  standard  resistance  of  0.1  megohm  or  more.  We  proceed 
to  a  description  of  these  various  ways  of  measuring  a  high  resist- 
ance and  the  precautions  which  should  be  observed. 

801.  Wheatstone-Bridge  Method  of  Measuring  a  Resistance 
of  from  10  to  1000  Megohms. — For  this  measurement  there  will 


x  l  R  =105 

> — vwvwwvv — 6 — vwvwwvw — < 


E  =  100  Volts 


O VWWWWWV  /WWNMMMA O 

6 
FIG.  801. 

be  required  a  standard  0.1  megohm  and  two  accurate  adjustable 
rheostats,  each  of  10,000  ohms  total  resistance;  a  D' Arson  val 
galvanometer  of  high  resistance  with  a  sensibility  of  109  megohms 
or  10~9  ampere;  and  a  source  of  direct  E.M.F.  of  about  100  volts. 
Make  the  Wheatstone-bridge  connections  as  shown  in  Fig.  801. 
Then  if  either  r  or  7*1  be  varied  until  the  galvanometer  shows  no 
deflection 

-r?«  (D 


154  MEASURING  ELECTRICAL  RESISTANCE         [ART.  801 

Suppose  the  standard  R  is  105  ohms,  and  r  is  plugged  at  its  ex- 
treme value  104  ohms  and  n  is  plugged  at  1  ohm,  then 

104 

x  =  -r-  105  =  109  ohms,  or  1000  megohms 
•  1 

is  the  greatest  resistance  which  can  be  measured  with  this  dis- 
position of  resistances.  If  R  is  a  megohm  then  ten  times  this 
resistance  may  be  measured,  provided  the  galvanometer  is  suffi- 
ciently sensitive. 

Suppose  the  galvanometer  will  deflect  one  scale  division  with 
10~9  ampere,  or  one  tenth  of  a  scale  division  with  10~10  ampere, 
and  we  wish  to  measure  x  to  within  1  part  in  1000,  when  x  =  109 
ohms  and  R  =  105  ohms.  We  have  to  inquire  what  E.M.F.  E 
must  be  applied  to  the  bridge  at  the  points  a  and  6.  The 
approximate  value  of  this  E.M.F.  is  easily  found  as  follows: 
First,  suppose  the  bridge  is  balanced  when  the  unknown  resist- 
ance has  the  value  x,  then  very  approximately,  the  current  i 

W 

which  will  flow  thru  x  is,  i  =  —  • 

The  fall  of  potential  over  x  with  the  bridge  balanced  is 

E  =  ix,  (2) 

and,  if  x  receives  an  increment  dx  this  fall  of  potential  is 

E  +  dE  =  ix  +  i  dx.  (3) 

Hence  dE  =  i  dx,  or 

dE  =  -dx.  (4) 


Now  dE  is  the  E.M.F.  effective  to  send  a  current  5i  thru  the 

*Tfi 

galvanometer  and  this  is,  very  approximately,  5i  =  —  .     Hence 

x 

dE  =  x  di  which  value  of  dE  put  in  Eq.  (4)  gives 


*-*  (5) 

Since  x  =  109  ohms,  di  =  10~10  ampere  and  dx  =  106  ohms,  for 
an  accuracy  of  1  part  in  1000,  we  obtain 

in-io 
E  =  1018  X  —jp  =  100  volts. 

One  hundred  volts  then  is  the  necessary  E.M.F.  for  measuring 
1000  megohms,  by  the  bridge  method,  to  an  accuracy  of  0.1  of  1 
per  cent  with  a  galvanometer  which  will  give  one  division  deflec- 
tion with  10~9  ampere. 


ART.  802]     THE  MEASUREMENT  OF  HIGH  RESISTANCE         155 

The  chief  precaution  to  observe  in  applying  this  method  is  to 
make  sure  that  the  galvanometer  and  the  0.1  megohm  are  per- 
fectly insulated  from  earth.  This  is  easily  accomplished  by 
setting  the  apparatus  upon  plates  of  glass  or  hard  rubber  and 
running  the  connections  thru  the  air,  supporting  the  wires  on 
glass  or  hard  rubber. 

It  is  well  also  to  include  in  the  battery  circuit  a  resistance  of 
not  less  than  1000  ohms  to  protect  the  galvanometer  from  injury, 
should  the  resistance  being  measured  break  down. 

802.  Use  of  a  Capacity  in  Connection  with  a  Wheatstone 
Bridge  for  High-Resistance  Measurements.  —  This  method, 


FIG.  802. 


which  is  taken  from  "  Measurement  of  Electrical  Resistance," 
by  W.  A.  Price,  is  added  for  the  sake  of  completeness.  The 
author  thinks,  however,  there  would  not  be  much  occasion  for  its 
employment  in  view  of  the  better  methods  which  are  available. 

The  diagram,  Fig.  802,  is  almost  self-explanatory. 

Here  a  condenser  C  is  charged  to  the  difference  of  potential 
of  the  points  a,  6  by  moving  the  levers  of  the  key  so  lever  q  is 
insulated  and  lever  p  makes  connection  with  point  1,  and  then 
the  condenser  is  discharged  thru  the  galvanometer  Ga  by  moving 
the  levers  so  p  is  insulated  and  q  makes  connection  with  point  2. 

The  resistance  r  or  rx  is  adjusted  until,  when  the  condenser  is 
connected  for  discharge,  there  is  no  deflection.  When  this  adjust- 


156 


MEASURING  ELECTRICAL  RESISTANCE         [ART.  803 


ment  is  effected,  a  and  6  must  be  at  the  same  potential  and  the 
ordinary  Wheatstone-bridge  formula  holds,  giving 


X  =  -R. 


(1) 


The  advantage  which  the  method  is  supposed  to  possess  is  one  of 
greater  sensitiveness,  which  results  from  the  fact  that  C  has 
time  to  become  fully  charged  by  the  slow  leak  of  current  thru  X, 
and  this  charge  is  then  able  to  expend  its  energy  suddenly  upon 
the  galvanometer,  producing  a  deflection  far  greater  than  would 
be  obtained  by  the  same  degree  of  unbalance  of  the  bridge  and 
with  the  galvanometer  joined  directly  to  the  points  a  and  b.  In 
using  the  method  the  condenser  C  should  be  chosen  of  as  great 
capacity  as  possible  and  the  galvanometer  should  be  of  very  high 
resistance. 

803.   Major  Cardew's  Electrometer  Method  of  Measuring  a 
High  Resistance.  —  In  this  method,  proposed  by  Major  Cardew, 


FIG.  803. 

(Fig.  803)  the  standard  R  which  must  be  made  variable  and  the 
high  resistance  X  to  be  measured  are  joined  in  series  and  their  free 
ends  a  and  b  are  connected  to  the  quadrants  of  an  electrometer. 
The  vane  is  joined  to  the  point  of  junction  c  of  the  two  resist- 
ances. The  resistance  R  is  varied  until  the  electrometer  shows 
no  deflection.  Then  X  =  R.  In  using  this  method  it  would  not 
be  necessary  to  produce  an  exact  balance.  The  deflection  of  the 
electrometer  is  noted  when  R  is  too  small,  and  calling  R  the  re- 
sistance and  d  the  deflection  we  then  increase  R  by  an  amount  5R 
and  again  note  the  deflection  d'  which  should  be  in  the  opposite 


ART.  806]      THE  MEASUREMENT   OF  HIGH  RESISTANCE         157 

direction.     If  for  small  deflections  we  assume  the  deflections  to 
be  proportional  to  the  potential  applied  to  the  vane,  we  have 


804.  The  Measurement  of  High  Resistances,  Unassociated 
with    an    Appreciable    Capacity  ;     Deflection    Methods.  —  The 

measurement  of  the  specific  resistances  of  insulating  materials, 
the  insulation  resistance  of  electrical  apparatus,  etc.,  is  not  a 
measurement  which  usually  demands  high  precision.  The  resis- 
tance of  insulating  materials  is  subject  to  considerable  fluctuation 
from  temperature  changes  and  other  causes  and  hence  the  less 
precise,  but  more  convenient  and  sensitive  deflection  methods 
are  to  be  preferred  to  the  null  methods  which  are  so  superior  in 
the  case  of  medium  and  low  resistances. 

In  describing  these  methods  we  shall  reserve  for  separate 
paragraphs  the  methods  of  measuring  the  insulation  of  cables  and 
condensers,  as  the  presence  of  an  appreciable  capacity  must  con- 
siderably modify  the  procedure. 

805.  The  Galvanometer  and  Accessory  Apparatus  for  High- 
Resistance    Measurement.  —  The   instrument   most   used    and 
best  adapted  to  very  high-resistance  measurements  by  a  deflec- 
tion method  is  the  galvanometer.     In  connection  with  the  galva- 
nometer a  standard  resistance  is  required.      This  standard  may 
be  either  a  megohm  or  a  one-tenth  megohm.     Because  of  the 
expense  of  the  former  the  latter  is  now  almost  universally  employed. 
To  increase  the  range  of  measurement  the  galvanometer  is  gen- 
erally used  with  a  high-resistance  shunt  which  is  made  variable. 
This  shunt  serves  the  same  purpose  in  high-resistance  measure- 
ments as  the  variable  ratio  arms  of  a  Wheatstone  bridge  in  the 
measurement   of   medium   resistances.     Special  types   of  highly 
insulated   keys   and   insulating   posts   and   plates   complete  the 
accessories  required.     We  proceed  to  give  the  theory  and  uses 
of  galvanometer  shunts. 

806.  Galvanometer  Shunts.  —  There  are  two    types  of  gal- 
vanometer shunts,  the  ordinary  and  the  universal  or  Ayrton  shunt. 

In  the  use  of  the  ordinary  shunt  the  resistance  of  the  galvanom- 
eter must  be  known.  In  the  diagram,  Fig.  806a,  let  S  be  the 
resistance  of  the  shunt  and  g  the  resistance  of  the  galvanometer. 
The  main  current  C  will  divide  thru  the  shunt  and  galvanometer 


S 

-^A/VWWNMMW 

C8 


158  MEASURING  ELECTRICAL  RESISTANCE         [ART.  806 

in  the  inverse  ratio  of  their  resistances.     If  C8  is  the  current  in 
the  shunt  and  Cg  the  current  in  the  galvanometer, 

— ?  —  —       and       C   4-  C    —  C 

C-  ,  d-IlU.  O0    T    U8     —    U, 

»      y 

whence, 

C'  =  j^SC  (D 

is  the   current   thru   the   galvanometer, 
and 

C.-^-sC  (2) 


Circuit 

is  the  current  thru  the  shunt. 

To  obtain  the  value  of  the  main  current  from  the  current  thru 
the  galvanometer,  we  have  from  Eq.  (1) 

„     I      O 

(3) 


The  quantity  M  =  ^—^  —  is  called  the  multiplying  power  of  the 

o 

shunt,  namely,  it  is  the  quantity  by  which  the   galvanometer 
current  must  be  multiplied  to  obtain  the  main  current. 

If  we  wish  to  make  the  current  in  the  galvanometer   -      of  the 


main  current,  that  is,  to  reduce  the  sensibility  of  the  galvanometer 

to   TTJ:  we  must  make  S  =  ,,      .,  • 
M  M  —  I 

For  putting  this  value  of  S  in  Eq.  (1)  we  have 


M-l 

The  introduction  of  the  shunt,  however,  changes  the  resistance 
of  the  circuit.  After  the  galvanometer  is  shunted  its  resistance 
will  be 


If  it  is  necessary  to  keep  the  resistance  of  the  circuit  constant, 


ART.  806]      THE  MEASUREMENT   OF  HIGH  RESISTANCE         159 


when  a  shunt  is  added  to  the  galvanometer,  there  must  be  intro- 
duced into  the  circuit  a  resistance  which  is 


M-l 

M 


(6) 


It  is  customary  to  make  shunt  boxes  so  that  M  may  be  given 
such  values  as  1,  10,  100,  1000,  and  10,000.     We  should  then  have 

Mi  =  1  S  =  oo 


M3  =  100 


=  1000 
=  10,000 


s  = 


9999 


Galvanometer 


Some  shunt  boxes  are  provided  also  with  means  for  adding  the 
proper  resistance  in  series  with  the  circuit  to  maintain  the  resist- 
ance of  the  circuit  constant  when  differ- 
ent values  are  given  to  the  shunt.  One 
arrangement  used  is  shown  in  Fig.  806b. 

Here  the  shunts  are  the  coils  Si,  S2, 
S$  and  the  compensating  resistances  the 
coils  0i,  02,  03,  and  04.  A  plug  inserted 
at  d  puts  the  circuit  directly  to  the  gal- 
vanometer without  a  shunt.  A  plug  in- 
serted at  b  and  &',  for  example,  shunts 
the  galvanometer  with  the  shunt  $2  and 
puts  into  the  circuit  the  compensating 
resistances  0i  +  02,  and  similarly  for 
plugs  inserted  at  a,  a'  or  c,  c' '.  A  plug 
at  e  short  circuits  the  galvanometer  and  puts  into  the  circuit  the 
compensating  resistance  0i  +  02  +  03  +  04,  which  sum  equals  the 
resistance  of  the  galvanometer  alone. 

This  type  of  shunt  should  be  wound  with  wire  of  the  same  tem- 
perature coefficient  as  the  wire  with  which  the  galvanometer  is 
wound.  Practically  all  galvanometers  are  wound  with  copper 


/  Circuit 


FIG.  806b. 


160 


MEASURING  ELECTRICAL  RESISTANCE         [ART.  807 


wire  and  this  changes  in  resistance  about  4  per  cent  for  every  10°  C. 
change  in  temperature.  Unless  the  coils  in  the  shunt  have  the 
same  temperature  coefficient  and  are  maintained  at  the  same  tem- 
perature as  the  coil  in  the  galvanometer  (a  matter  hard  to  realize 
in  practice)  unallowable  errors  may  result  from  the  employment 
of  this  type  of  shunt.  Furthermore  every  shunt  must  be  adapted 
to  the  particular  galvanometer  with  which  it  is  to  be  used. 
These  disadvantages  are  overcome  in  the  universal  or  Ayrton 
shunt,  which  is  the  kind  now  almost  universally  in  use.  The 
theory  and  use  of  the  Ayrton  shunt  is  as  follows : 

807.  The  Ayrton  or  Universal  Shunt.  —  Fig.  807a  shows  the 
disposition  of  the  circuits  employed,  a,  b,  c,  d,  e,  are  resist- 
ance coils  of  manganin  or  other  low-temperature-coefficient  wire. 
These  are  joined  in  series  and  the  galvanometer  terminals  are 
permanently  connected  to  the  terminals  of  the  series.  One  ter- 
minal of  the  main  circuit  is  permanently  joined  at  one  end,  as  at 


C2 

*•    a      2      b      3  _    c      4     _d     5      e 

P~< 


'AAAAA 


FIG.  807a. 

the  point  6,  and  an  arrangement  is  provided  by  which  the  other 
terminal  p  may  be  moved  to  any  of  the  points  1,  2,  3,  4,  5,  6. 
R  represents  the  total  resistance  and  E  the  E.M.F.  included  in  the 
main  circuit.     The  resistance  of  the  galvanometer  alone  is  g. 
It  was  shown,  Eq.  (3),  par.  806,  that  the  multiplying  power  of 

any  shunt  is  M  =  ^—^ — ,  where  S  is  the  total  resistance  of  the 

>o 

shunt  and  g  is  the  resistance  of  the  galvanometer.  We  can  now 
construct  for  the  different  resistances  in  the  galvanometer  circuit 
and  for  the  different  resistances  which  shunt  the  galvanometer, 
when  p  is  moved  from  the  point  1  to  2  to  3,  etc.,  the  following 
table : 


ART.  807]      THE   MEASUREMENT   OP  HIGH   RESISTANCE         161 


p  on  point 

Res.  in  gal.  circuit 

Value  of  shunt,  S 

Multiplying  power  of  shunt 

1 

g 

a+b+c+d+e 

g+a+b+c+d+e 
1        a+b+c+d+e 

2 

g+a 

b+c+d+e 

,,      g+a+b+c+d+e 

*•-       6+c+d+e 

3 

g+a+b 

c+d+e 

„,      0+a+fc+c+rf+e 

c+d+e 

4 

g+a+b+c 

d+e 

^      ^+a+6+c+rf+e 

4                rf+e 

5 

g+a+b+c+d 

e 

M      ^+a+6+c+d+e  ' 

e 

6 

g+a+b+c+d+e 

0 

M6  =  Infinity. 

It  now  appears  from  the  4th  column  of  this  table  that  the 
relative  value  of  the  multiplying  power  of  the  shunt  for  any  two 
positions  of  the  contact  p  is  independent  of  the  resistance  of  the 
galvanometer.  Thus,  calling  a+b+c+d+e=r 


Mi 

Mi 
M4 
Mi 
M3 


b+c+d+e 


M2 
M!_ 


It  follows  that,  if  the  resistances  are  chosen  so  e  =  0.0001  r, 
d  +  e  =  0.001  r,  c  +  d  +  e  =  0.01  r  and  6  +  c  +  d  +  e  =  0.1  r, 
the  sensibility  possessed  by  the  galvanometer  (when  shunted  with 
the  resistance  r)  will  become  0.1,  0.01,  0.001,  or  0.0001  as  great 
according  as  the  contact  p  rests  on  point  2,  .3,  4,  or  5.  This 
result  is  obtained  theoretically  with  a  galvanometer  of  any  resis- 
tance and  with  any  value  given  to  the  total  resistance  r.  The 
question  then  arises:  what  considerations  govern  the  value  which 
should  be  given  to  r?  It  will  be  observed,  if  the  resistance  r 
is  made  very  high  as  compared  with  the  resistance  of  the  galva- 
nometer, that,  with  the  contact  on  point  3  or  4,  the  galvanometer 
has  thrown  in  series  with  it  a  very  considerable  resistance  which 
will  reduce  greatly  the  current  C  in  the  main  circuit  (Fig.  807a) 
unless  the  resistance  R  in  this  main  circuit  is  also  very  high.  On 
the  other  hand,  if  the  resistance  r  is  made  very  small,  as  com- 
pared with  g,  the  galvanometer  being  permanently  shunted  with 
a  low  resistance  has  its  intrinsic  sensibility  much  reduced.  Also, 


162  MEASURING  ELECTRICAL  RESISTANCE         [ART.  807 

if  this  is  a  D'  Arson  val  galvanometer  it  will  be  overdamped 
when  r  is  small,  and  the  coil  will  move  sluggishly.  Experience 
and  practice  show  that  r  should  be  chosen  approximately  ten  times 
the  average  resistance  of  the  galvanometers  which  are  to  be  used 
with  the  shunt. 

In  considering  the  principle  of  the  Ayrton  shunt  it  should  be 
carefully  noted,  that,  while  the  shunt  reduces  the  sensibility  of 
any  galvanometer  in  a  definite  way  it  does  not  in  general  reduce 
the  current  thru  the  galvanometer  in  the  same  definite  way.  Thus, 
if  we  call  C'  the  main  current  when  p  (Fig.  807a)  is  on  point  1, 
the  galvanometer  current  will  be 

r  Cf 

Cf  _        '       ni  _ 
a    —         i         ^     —    TI  /r   ' 

g  +  r  Mi 

If  the  contact  is  now  moved  to  some  other  point  as  3,  and  we  call 
the  main  current  which  is  then  flowing  C'"  the  galvanometer 
current  will  be 

,„      c  +  d  +  e 


_ 
g  +  r  ~  Ms  ' 

The  ratio  of  the  galvanometer  currents  in  these  two  cases  is 


C0r  '  r          C'       M3   C' 

Only  when  the  external  resistance  R  is  very  large,  so  that  the 
effective  resistance  of  the  entire  circuit  remains  practically  con- 
stant for  the  different  positions  of  the  shunt  contact,  will  the 
current  C"'  be  the  same  as  the  current  C'.  In  this  case  only  will 
the  ratio  of  the  galvanometer  currents  for  any  two  positions  of 
the  shunt  contact  be  in  the  inverse  ratio  of  the  multiplying  powers 
of  the  shunt  for  these  two  positions. 

If  by  any  device  the  main  current  C  is  maintained  exactly 
constant,  then  the  shunt  will  exactly  cut  down  the  galvanometer 
current  in  the  same  way  it  cuts  down  the  sensibility.  In  measur- 
ing insulation  resistances,  R  is  usually  very  large  and  the  Ayrton 
shunt  may  be  used  not  only  as  a  device  to  reduce  galvanometer 
sensibility  but  also  to  reduce  in  like  manner  the  galvanometer 
current  by  known  and  fixed  amounts.  It  is  in  this  latter  way  and 
for  this  purpose  that  the  Ayrton  shunt  is  chiefly  used  and  it  is 
therefore  necessary  to  investigate  the  magnitude  of  the  errors 
introduced,  under  different  conditions  of  use,  when  it  is  assumed 


ART.  807]      THE   MEASUREMENT   OF  HIGH  RESISTANCE         163 

that  the  galvanometer  current  is  cut  down  in  the  same  proportion 
as  the  galvanometer  sensibility. 

Referring  to  Fig.  807a,  the  current  thru  the  galvanometer  is 

c*  =  jic>  a) 

where  M  is  the  multiplying  power  of  the  shunt   (which  takes 
values  MI,  M2,  etc.,  according  as  p  is  on  point  1,  2,  etc.). 

Call  Ra  the  shunted  value  of  the  galvanometer  resistance  and 
R  the  resistance  in  the  main  circuit.  Then  if  E  is  the  E.M.F. 
of  the  source, 

c-      E 
R  +  R, 

and  c  "      E  < 


Let  the  contact  be  upon  a  point  p  such  that  M  =  Mp  and 
Rs  =  R8f,  then  the  galvanometer  current  will  be 


°  p  R+R,' 

Now  move  the  contact  to  a  point  q  such  that  M  =  Mq  and 
R,  =  R8"',  then  the  galvanometer  current  will  be 

r  "  =  —        1  u\ 

Mq  R+Ra" 

By  taking  the  ratio  of  Eq.  (4)  to  Eq.  (3)  we  find 

(5) 


C0"      Mp  R+R8' 


C0'      MqR  +  Ra' 

-p    I     -p  r 

Eq.  (5)  shows  that,  in  so  far  as  p   ,      *,  differs  from  unity  the 

rt  +  Ha 

ratio  of  the  galvanometer  currents  for  any  two  positions  of  the 
shunt  differs  from  the  inverse  ratio  of  the  multiplying  powers  of 
the  shunt  for  the  two  positions. 
Since 

1  -1-  Ra' 
R+R8f  ~R 

R  +  R." 


we  note  that  when  R  is  very  large  the  fraction  is  practically  unity, 
that  is,  the  current  in  the  main  circuit  is  practically  constant, 
while  the  current  in  the  galvanometer  is  changed  in  the  same  ratio 
but  inversely  as  the  multiplying  power  of  the  shunt  is  changed. 


164 


MEASURING  ELECTRICAL  RESISTANCE         [ART.  807 


The  following  typical  case  will  serve  to  show  the  magnitude  of 
the  errors  which  actually  result  from  assuming  that  the  current 

in  the  galvanometer  is  altered  in  the  ratio  -—•  -     Reference  is  here 

Mq 

made  to  Fig.  807b. 


— .Olr ; 

— .OOlr- 

k-.OOOlr- 


By  means  of  the  handle  H  the  contact  can  be  moved  to  positions 
1,  0.1,  0.01,  0.001,  0.0001,  0,  Inf.  In  the  case  selected  the  shunt 
is  intended  for  use  with  galvanometers  having  from  100  to  500 
ohms  resistance.  We  shall  take 

g  =  350  ohms, 
r  =  3000  ohms, 
and  first  assume  that  the  external  resistance  is 

R  =  105  ohms. 
Then 


a+b 


=  r  =  3000  ohms  and  M1  = 


b  +  c  +  d  +  e  =  0.1  r  =    300  ohms  and  M2  =  10  Mi, 


c+d+e=  0.01  r  = 

d  +  e  =  0.001  r= 

e  =  0.0001  r  = 


30  ohms  and  M3  =  100  Mi, 
3  ohms  and  M4  =  1000  MI, 
0.3  ohm    and  M5  =  10,000  MI. 


ART.  807]     THE  MEASUREMENT  OF  HIGH  RESISTANCE         165 


The  shunted  galvanometer  resistance  then  takes  the  following 
values  : 

gr         350  X  3000 

R,1  =  —  -  =  -  ^^  -  =  313  ohms. 
g  +  r  3350 


= 


p  In  _ 


g  +  r 


(350  +  2700)  X  300      QnQ 
—  -  oorrt  -  =  oUo.U, 
3350 


g  +  r 


(350  +  2700  +  270)  X  30 
3350 


_  OQ  7Q 


IV  _  (g+a+b+c)(d+e)  _  (350  +2700  +270  +27)  X  3  _     QQ7 
r  3350 


p  v_ 


a  +  b  +  c+d)e  _  (350  +  2700  +  270  +  27  +  2.7)  X  0.3 
g  +  r  3350 

=  0.3. 

If  we  now  put  these  numerical  values  in  expressions  of  the  form 
given  in  Eq.  (5),  we  obtain  the  following  values  for  the  current 
in  the  galvanometer  with  the  shunt  in  positions  1,  2,  3,  4,  5: 


For  position 

Galvanometer  current 

1 

C  n     C?  v  1Q5+313 

"         10  X  105+303 
rm     C,i      105+313 

100  A  105+30 
r  TV      Cffi      105+313 

K. 

*        1000  X    1Q5+3 

Cff  ~  10000  X  1Q5+0.3 

In  this  case  it  is  seen  that  the  last  terms  differ  very  little  from 
unity.     Thus  the  largest  departure  is  for  the  position  5,  where 

IP  4- 313 


105  +  0.3 


=  1.00313. 


With  an  external  resistance  as  great  as  100,000  ohms  and  a  shunt 
of  total  resistance  3000  ohms  it  is  legitimate  to  assume,  for  most 
work,  that  the  shunt  cuts  down  the  galvanometer  current  in  the 
same  way  as  it  cuts  down  the  galvanometer  sensibility,  that  is, 
inversely  as  the  multiplying  power  of  the  shunt.  If,  on  the  other 
hand,  R  is  made  as  low  as  1000  ohms  the  error  for  position  5  of 
the  shunt  would  be  as  much  as  31.3  per  cent. 


166  MEASURING  ELECTRICAL  RESISTANCE         [ART.  808 

We  can  now  find  from  the  galvanometer  deflection,  with  the 
shunt  set  in  any  position,  the  value  of  the  main  current  C  as 
follows:  By  the  principle  embodied  in  Eq.  (1),  par.  807,  the  cur- 
rent thru  the  galvanometer  is  equal  to  the  main  current  divided 
by  the  multiplying  power  of  the  shunt,  or,  in  general, 

Cr» 
.  ,  O        r 

L°~M  = 


808.  Galvanometer  Constant,  Obtained  by  Using  an  Ayrton 
Shunt.  —  To  determine  the  constant  of  the  galvanometer  using 
an  Ayrton  shunt  to  which  the  galvanometer  is  permanently 
attached  (as  in  Fig.  807b)  we  may  proceed  as  follows: 

Let  C'  =  MiCg'  be  the  main  current  with  the  shunt  in  position  1  , 
C"  =  M^Cg"  be  the  main  current  with  the  shunt  in  position  2, 
C'"  =  MzCgf  be  the  main  current  with  the  shunt  in  position 
3,  with  similar  expressions  for  the  other  shunt  positions. 

If  the  galvanometer  current  is  Cg'  =  K\  d\  for  the  shunt  in 
position  1,  where  K\  is  a  constant  and  d\  the  galvanometer  deflec- 
tion, we  have 

C'  =  M&tdi 


where  K  is  another  constant.     Similarly 
C"  =  M2#i  dz  =  10  M&i  d2  =  10 


C'"  =  100Kd3,etc. 
Also, 


where  JR.',  R8",  Rs'n ',  etc.,  are  the  shunted  galvanometer  resistances 
with  the  contact  on  positions  1,  2,  3,  etc.     We  therefore  obtain 

E 
R  +  Raf==Kdl' 

E        1 
or  K  =  -T-, 

JK  -f-  ria    Cti 

or          K  =  0.1R_^Rf,    j-,    or    K  =  0.01      _^    ,„    J-etc. 
Or,  in  general, 


Where  N  refers  to  the  numbers  1,  0.1,  0.01,  etc.,  stamped  upon  the 


ART.  809]      THE   MEASUREMENT   OF   HIGH   RESISTANCE         167 

shunt  for  the  shunt  position  used,  Ra  is  the  shunted  galvanometer 
resistance  for  that  position  and  d  is  the  galvanometer  deflection. 
In  determining  the  constant  K  it  is  customary  to  take  R  so  large, 
usually  105  ohms,  that  R8  is  negligible  in  comparison.  For  insula- 
tion testing  this  approximation  may  be  permitted  and  we  have, 
with  sufficient  precision  for  many  purposes, 

NE 


809.   Insulation  Measurements  with  a  Galvanometer  and  an 
Ayrton  Shunt.  —  When  the  insulation  resistance  to  be  measured 


Galv; 


Ayrton 
Shunt 


Flexible  Cord 


Battery 


100,000  Ohm 
Box 


FIG.  809. 

has  a  comparatively  small  capacity  and  dielectric  absorption  the 
procedure  is  very  simple  and  is  carried  out  in  practice  as  follows: 

A  galvanometer  (usually  a  D'Arsonval  instrument),  a  standard 
one-hundred-thousand-ohm  box,  an  Ayrton  shunt,  and  the  resist- 
ance to  be  measured  are  joined  as  in  Fig.  809. 

In  a  convenient  type  of  construction,  the  battery  key  6  is 
combined  with  the  Ayrton  shunt  for  compactness  and  also  for 
convenience  in  manipulation.  The  handle  a  controls  the  key  b 


168  MEASURING  ELECTRICAL  RESISTANCE         [ART.  809 

and  is  mounted  so  that  it  projects  up  thru  the  handle  which 
operates  the  shunt.  By  depressing  a,  the  contact  b  is  closed. 
This  is  arranged  so  that  it  can  be  locked  in  the  closed  position. 
If  the  shunt  and  the  D'Arsonval  galvanometer  are  properly 
related  the  latter  will  be  just  aperiodic. 

The  shunt  may  be  held  in  one  hand  while  its  handle  and  key  is 
manipulated  with  the  other  hand.  Both  the  shunt  and  one- 
hundred-thousand-ohm  box  should  be  highly  insulated  and  are, 
for  this  purpose,  often  constructed  entirely  of  hard  rubber. 

To  make  a  measurement  of  insulation  resistance,  of  a  short 
length  of  cable  for  example,  the  procedure  would  be  as  follows:  A 
battery  of  50  or  100  dry  cells  is  used.  These  should  be  first  joined 
directly  to  the  one-hundred-thousand-ohm  box  instead  of  to  the 
cable  as  indicated  in  the  figure.  The  constant  of  the  galvanometer 
may  now  be  obtained.  This  is  not  the  same  constant  as  that  given 
by  Eq.  (2)  in  par.  808,  which  is  the  true  galvanometer  constant. 

It  is  an   arbitrary  constant  defined  by  the   relation  D  =  -^—  . 

Here  G  is  the  constant  sought,  N  the  shunt  setting  0.1,  0.01,  0.001, 
or  0.0001.  The  0.1  is  the  one  hundred  thousand  ohms  expressed 
in  megohms,  and  D  is  the  galvanometer  deflection  which  is 
obtained  with  the  particular  battery  used  for  the  measurement. 
Thus  we  have 

G=IQN'  (1) 

In  obtaining  this  constant,  the  shunt  is  first  set  at  zero.  The 
battery  circuit  is  then  closed  by  depressing  the  battery  key  b 
and  the  shunt  is  moved,  first  to  N  =  0.0001,  and  the  deflection 
no-ted.  If  this  is  less  than  25  small  scale-divisions,  the  shunt  is 
moved  to  N  =  0.001,  and,  if  still  less  than  25  scale-divisions,  to 
N  =  0.01.  The  final  deflection,  which  should  not  be  less  than 
25  divisions,  is  noted  and  called  D.  The  shunt  setting  also  being 
noted,  the  constant  is  given  by  expression  (1)  above. 

The  constant  G  having  been  thus  determined  the  insulation 
resistance  may  now  be  measured  by  the  following  procedure: 
First  connect  the  battery  to  the  cable  or  resistance  to  be  measured, 
as  shown  in  Fig.  809.  Reset  the  shunt  to  position  zero.  The  one- 
hundred-thousand-ohm  box,  or  0.1  megohm,  may  be  left  in  circuit 
or  it  may  be  short-circuited.  In  the  former  case  the  resistance 
measured  will  include  this  and  will  be  0.1  megohm  too  large. 


ART.  809]      THE   MEASUREMENT  OF  HIGH  RESISTANCE         169 

Close  the  battery  key  b.  If  the  cable  has  any  considerable  capac- 
ity and  dielectric  absorption,  sufficient  time  must  be  allowed  for 
the  cable  to  become  fully  charged.  No  definite  time  for  this  can 
be  specified  and  this  matter  will  receive  further  treatment  later 
on.  But  assuming  electrification  is  complete,  move  the  shunt 
successively  to  positions  0.0001,  0.001,  0.01,  etc.,  until  the  deflec- 
tion obtained  is  as  large  as  possible  and  yet  remains  upon  the 
galvanometer  scale.  Note  this  deflection  calling  it  d,  and  also 
the  shunt  position  used  and  call  it  Ni.  Then  in  the  same  way  that 
we  obtained 


where  /  is  the  insulation  resistance  sought  expressed  in  megohms, 
or 


The  above  procedure  assumes  that  the  insulation  resistance  of 
the  lead  wires  and  the  one-hundred-thousand-ohm  box  is  infinite. 
This  may  not  be  the  case,  however,  and  the  matter  must  be  tested 
by  disconnecting  the  wire  from  the  core  of  the  cable  and  noting  if 
there  is  any  deflection,  the  shunt  setting  remaining  the  same. 
If  there  is  a  small  deflection  di  it  must  be  subtracted  from  d,  and 
then  the  final  expression  for  the  true  insulation  resistance  becomes 

T          GNl  (V 

*"J=Tk' 

It  is  the  insulation  resistance  of  the  entire  cable.     If  its  total 
length  is  L  feet,  then  its  insulation  resistance  per  mile  will  be 

7  G^L  (4) 

(d  -  d^  5280 

The  following  example  will  illustrate  the  use  of  formulas  (1)  and  (4)  : 
The  galvanometer  deflection,  in  obtaining  its  constant,  was 
D  =  83  millimeter-divisions.     The  shunt  setting  was  N  =  0.0001. 
Hence  the  constant  was 

CO 

G  =  =  83'000- 


This  galvanometer  was  then  used  to  determine  the  insulation 
resistance  per  mile  of  a  cable  3200  feet  long.  The  deflection  due 
to  the  cable  and  lead  wires  was  d  =  53  divisions,  and  the  shunt 


170  MEASURING  ELECTRICAL  RESISTANCE         [ART.  810 

position  used  was  Ni  =  1.  The  deflection  due  to  the  lead  wires 
alone  was  di  =  3  divisions.  Hence  by  Eq.  (4) 

83,000  XIX  3200 
Im  =     (53  -  3)  X  5280     =  10°6  meg°hms  per  mile' 

8 10.  Measurement  of  High  Resistances  by  Leakage  Methods. 
-  When  a  high  resistance  is  practically  free   from  capacity  it 

may  be  quite  accurately  and  easily  measured  by  observing  the 
time  which  is  required  for  a  certain  portion  of  the  charge  in  a 
condenser  to  leak  thru  the  high  resistance.  If  the  high  resistance 
is  associated  with  an  appreciable  capacity,  as  is  the  case  with  the 
insulation  resistance  of  a  cable,  the  methods  of  leakage  may  still 
be  applied  but  require  certain  modifications.  The  leakage  methods 
which  follow  apply  when  the  capacity  associated  with  the  resist- 
ance is  negligible. 

811.  Theory  of  Leakage  of  Condensers.  —  Assume,  as  a  first 
approximation,  that  the  resistance  from  one  terminal  to  the  other 
of  the  condenser  itself  is  infinite. 

Call  C  the  capacity  .of  the  condenser.  First,  let  the  condenser 
be  charged  to  a  potential  V\.  Second,  join  the  terminals  of  the 
condenser  with  a  high  resistance  R  and  maintain  this  connection 
for  a  time  T  until  the  potential  of  the  condenser  falls  to  a  value 
Vi.  Then  disconnect  the  resistance  and  measure  the  potential 
Vi.  The  following  relations  will  now  hold: 

q  =  Cv, 

where  q  is  the  quantity  of  electricity  in  the  condenser  at  any 
instant  when  its  potential  is  v. 

.  _  _dq  _   _  p  dv 
~~dt~       "~dt 

is  the  current  which  leaves  the  condenser  when  the  potential 
changes  with  the  time.  With  the  resistance  R  as  the  only  resist- 
ance in  circuit  with  the  condenser  this  current  must  also  equal,  by 

Ohm's  law,  -5.     Hence, 

v  _        ~  dv  (^ 

R  dt 


or 


£T  rv*dv 

dt=-RC   I  (2) 

Jv1    v 


from  which  we  obtain,  by  integration, 

T  =  -RC  (log.  V,  -  loge 


ART.  812]     THE   MEASUREMENT   OF  HIGH  RESISTANCE         171 


or 


(3) 


In  Eq.  (3)  C  is  in  farads,  R  in  ohms  and  T  in  seconds.  Since  V\ 
and  Vz  occur  as  a  ratio  they  may  be  taken  in  any  units.  Express- 
ing C  in  microfarads  and  changing  to  common  logarithms,  we  have 


T  =  2.3026  X  10-6  RCf  logio  £• 


(4) 


Eq.  (4)  is  a  useful  relation  between  time,  resistance  and  capacity, 
which  enables  any  one  of  these  three  quantities  to  be  calculated 
when  the  other  two  have  been  determined. 

If  we  express  the  resistance  in  megohms  and  call  it  Rm,  we  have, 
within  0.018  of  1  per  cent, 

I?  T  ^ 

•fim  \"/ 

O  2.303  logw^-1 

812.  High  Resistance  Measured  by  Leakage;   Method  I. — 

In  this,  the  simplest  application  of  the  method  of  leakage,  the 
apparatus    required    is    a    bal- 
listic D'Arsonval  galvanometer, 
of     moderate     sensibility    and 
proportional  scale,  a  mica  con- 
denser (preferably  of  1  micro-     ^ 
farad  and  adjustable  in  steps  of 
0.05   microfarad)    and   suitable 
highly    insulated    keys.       The 
connections  may  be  made  as  in  .       i      a  T., 

Fig.  812a.  Kl 

In  the  figure,  Rm  is  the  resist- 
ance in  megohms  to  be  meas- 
ured, Cf  the  mica  condenser  ' 'I'l'l'l 

and  Ga  the  ballistic  galvanom-  Ba 

eter.     The  E.M.F.  of  the  bat-  FlG-  812a- 

tery  Ba  should  be  chosen  such  that  with  the  galvanometer  and 

condenser  C/,  which  are  used,  a  deflection  to  near  the  end  of 

the  galvanometer  scale  will  be  obtained  when  the  condenser  is 

charged  and  then  immediately  discharged  thru  the  galvanometer. 

To  measure  the  resistance  Rm  the  key  K\  is  first  closed  and 
then  opened,  and  immediately  thereafter  the  key  Kt  is  closed 
which  discharges  the  condenser  C/,  charged  to  the  full  potential 


172  MEASURING  ELECTRICAL  RESISTANCE         [ART.  812 

of  the  battery,  thru  the  ballistic  galvanometer.  The  deflection 
di  is  carefully  noted.  If  the  galvanometer  is  proportional  in  its 
readings  to  the  quantity  of  the  electricity  discharged  thru  it,  we 
shall  have  the  deflection  of  the  galvanometer, 


i  =  kV1} 

where  Qi  is  the  quantity  of  electricity  discharged,  K  and  k  are 
constants,  and  Vi  is  the  potential  of  the  battery. 

The  key  K2  is  left  open  and  the  key  KI  is  again  closed.  KI  is 
now  opened  for  a  known  time  T  seconds.  While  KI  and  K2  are 
open  the  condenser  C/  is  losing  its  charge  by  leakage  thru  the 
high  resistance  Rm.  At  the  end  of  T  seconds  K2  is  closed  and  the 
electricity  which  remains  in  the  condenser  is  discharged  thru  the 
galvanometer  giving  a  deflection  dz  =  KQ2  =  KCfV2  =  kV2. 

The  resistance  in  megohms  is  then  given,  as  in  Eq.  (5),  par. 
811,  by  the  relation 

T     -  CD 


The  result  expressed  by  Eq.  (1)  assumes  that  the  insulation  re- 
sistance of  the  condenser  itself  is  perfect.  If  the  condenser  leaks, 
as  will  generally  be  the  case,  a  separate  experiment  must  be  made 
to  determine  its  insulation  resistance.  To  do  this,  disconnect  the 
resistance  Rm  at  a,  and  proceed  in  exactly  the  same  manner  as 
above.  Let  Rmr  be  the  resistance  of  the  condenser  in  megohms 
which  is  found,  and  call  Rm"  the  resistance  of  the  condenser  and 
specimen  when  in  parallel.  Then 

J_    J_.  _L 

r>    //          r>        \     r>    n 
Km  Km          Km 

from  which  Rm=   pf^,,-  (2) 

K       —  Km 

It  is  to  be  noted,  that  the  choice  of  the  time  T  and  the  capacity 
Cf  in  Eq.  (1)  is  arbitrary,  and  the  question  arises,  How  should 
these  quantities  be  selected  in  order  that  the  precision  of  the 
measurement  shall  be  as  great  as  possible? 

We  may  consider  that  the  conditions  of  the  measurement  are 
made  as  favorable  to  precision  as  possible,  when  matters  are  so 
arranged  that  the  proportionate  error  in  the  time  being  measured 
is  equal  to  the  proportionate  error  in  the  deflection  being  read. 


AKT,  812]     THE   MEASUREMENT   OF  HIGH  RESISTANCE        173 

Now  the  throw  deflection,  after  the  leakage  has  taken  place,  is 
proportional  to  the  E.M.F.  to  which  the  condenser  is  then  charged, 
namely  to  72.  We  may  write,  then,  as  the  condition  for  maximum 

precision, 

ST         SV2 

¥     "TT 

The  negative  sign  is  here  used  because  as  the  time  increases,  the 
potential  decreases.  If  we  express  the  time,  as  in  par.  811,  by 
the  relation 

T  =  -  RC  loge  72  +  RC  loge  7 1 

and  call  7i  constant,  we  obtain 

,T          Kr8Vi  672       8T 

•BCTV°r"  T^flC' 
From  Eqs.  (3)  and  (4)  we  thus  obtain 

T  =  RC.  (5) 

Putting  T  =  RC  in  Eq.  (3),  par.  811,  we  obtain 


Since  the  throw  deflections  may  generally  be  taken  as  proportional 
to  the  potentials  7i  and  F2  we  have,  if  di  and  d2  are  these  deflec- 
tions, 

*  =  7  =  |f-  (6) 

(For  functions  of  e  see  Appendix  II,  1.) 
Further,  if,  in  Eq.  (3),  par.  811,  we  write  72  =  — ,  we  have 

6 

T  =  RC  loge  e,  or,  as  loge  e  =  1, 
T  =  RC. 

Thus,  if  we  make  the  second  deflection  -  of  the  first,  we  have 

6 

T 

R  =  -ft  and  the  calculation  is  simplified.     If  time  is  expressed  in 

seconds,  resistance  in  megohms  and  capacity  in  microfarads,  we 
have 

T  =  RmCf.  (7) 

Eq.  (7)  implies  that,  for  highest  precision,  the  time  in  seconds 
that  the   condenser  is  permitted   to  leak  has  been  so  chosen 


174 


MEASURING  ELECTRICAL  RESISTANCE         [ART.  812 


that  it  is  equal  to  the  product  of  the  resistance  being  measured 
expressed  in  megohms,  times  the  capacity  used,  expressed  in 
microfarads.  To  use  relation  (7)  we  should  first  choose  C/  of  any 
convenient  value  and  then  by  a  preliminary  trial  find  roughly  the 
value  of  Rm.  After  this  the  measurement  should  be  repeated 
allowing  the  condenser  to  leak  a  time  T  =  RmC/.  If  this  time  is 
inconveniently  long  or  short  then  the  capacity  C/  may  be  reduced 
or  increased  when  the  time  of  leak,  for  best  precision,  will  change 
in  the  same  proportion. 

In  ordinary  practice  it  is  not  very  essential  to  closely  regard 
the  above  rule  for  highest  precision,  and  the  conditions  for  pre- 
cision will  be  sufficiently  met  if  the  condenser  is  allowed  to  leak 
a  time  such  that  F2  is  about  one-half  or  one-third  of  V\.  If  the 

second  deflection  dz  is  made  -  =  ^ ,  or  37  per  cent  of  the  first 

C         Z, ,  i 

deflection  d\,  the  best  conditions  will  be  exactly  realized. 


FIG.  812b. 

The  above  procedure  and  arrangement  of  circuits  for  the  meas- 
urement of  resistance  by  leakage  may  be  modified  in  details  ii^  a 
variety  of  ways.  For  example,  for  the  keys  KI  and  K2,  one  may 
substitute  a  Rymer-Jones  key  when  the  connections  would  be  as 
shown  in  Fig.  812b. 

In  this  arrangement  the  first  operation  is  to  throw  the  handles 
of  the  Rymer-Jones  key  k  to  the  right  so  as  to  charge  the  con- 
denser to  the  difference  of  potential  of  the  battery  Ba.  As  soon  as 
it  is  charged,  the  left-hand  handle  of  the  key  is  thrown  to  the  left, 
when  the  condenser  begins  to  leak  thru  the  resistance  Rm.  When 
this  has  continued  for  T  seconds,  the  right-hand  handle  is  thrown 
to  the  left  which  discharges  the  remaining  electricity  thru  the  gal- 
vanometer. The  high  resistance  is  now  calculated  in  the  way 
given  above. 


ART.  813]     THE  MEASUREMENT   OF  HIGH  RESISTANCE         175 

813.   High  Resistance  Measured  by  Leakage;  Method  II.— 

An  electrometer  of  the  quadrant  type  may  be  substituted  for  a 
galvanometer.  In  this  case  the  connections  may  be  made  as  in 
Fig.  813. 

If  the  electrometer  does  not  give  deflections  proportional  to 
the  potential  applied  to  its  quadrants  it  is  necessary  to  make  a 
preliminary  calibration  of  its  scale,  plotting  in  a  curve  potentials 
against  deflections.  In  this  arrangement  the  needle  is  main- 
tained charged  by  the  dry  pile  or  other  source  of  high  potential  6. 


.Earth 


FIG.  813. 


The  specimen  Rm  is  connected  in  parallel  with  the  mica  con- 
denser Cf  the  opposite  sides  of  which  are  joined  to  the  two  pairs  of 
quadrants  of  the  electrometer.  By  means  of  the  battery  Ba  the 
condenser  and  the  electrometer  quadrants  are  charged  to  the 
same  potential  V\.  The  battery  Ba  is  then  disconnected  and 
the  time  accurately  noted  for  the  potential  to  fall  from  V\  to  V^} 
the  values  of  V\  and  V2  being  obtained  from  the  observed  deflec- 
tions which  will  give  the  potentials  from  the  previously  obtained 
curve. 

With  the  excellent  galvanometers,  having  nearly  proportional 
scales,  which  are  now  available,  one  would  hardly  select  for  this 
test  an  electrometer  in  preference  to  a  galvanometer. 

In  the  above  class  of  measurements  it  generally  happens  that 
the  specimen  Rm  being  measured  has  not  only  resistance  but 
more  or  less  capacity.  This  would  be  the  case  if  the  sample  were 
a  few  meters  of  rubber-covered  wire  of  which  the  insulation  resist- 
ance per  unit  of  length  of  the  covering  is  to  be  determined. 

When  such  is  the  case,  for  precision,  the  capacity  of  the  sample 
must  be  determined  by  a  preliminary  experiment.  As  this 


176  MEASURING  ELECTRICAL  RESISTANCE         [ART.  814 

capacity  will  be  in  shunt  to  the  condenser  C/,  used  in  the  test,  it 
must  be  added  to  the  capacity  of  C/  in  estimating  its  value.  This 
capacity  of  the  sample  may  usually  be  determined  with  sufficient 
exactness  by  charging  it  to  a  high  potential  and  immediately 
thereafter  discharging  it  thru  a  ballistic  galvanometer.  The 
throw  deflection  obtained  when  compared  with  the  throw  deflec- 
tion which  may  be  obtained  by  discharging  a  known  capacity 
thru  the  galvanometer,  will  give  the  value  of  the  unknown  capacity. 

If  the  capacity  of  the  sample  resistance  is  very  considerable 
it  will  then  not  be  necessary  to  use  any  auxiliary  condenser,  it 
being  only  necessary  to  note  the  time  of  leak  of  the  charge  of  the 
capacity,  associated  with  the  high  resistance  under  measurement. 
Such  a  case  arises  in  the  measurement  by  leakage  of  the  insulation 
resistance  of  a  submarine  cable. 

814.  Insulation  Resistance  of  a  Celluloid  Condenser  Obtained 
by  the  Method  of  Leakage.  —  We  now  give  a  description  of  an 
actual  determination,  by  the  method  of  leakage,  of  the  insulation 
resistance  of  a  celluloid  condenser.  It  will  serve  to  illustrate 
what  has  been  said  above  respecting  the  determination  of  high 
resistance  by  leakage  and  will  bring  out  some  of  the  peculiarities 
of  celluloid  and  the  difficulty  of  precisely  defining  what  constitutes 
the  true  ohmic  resistance  of  a  material  of  this  nature. 

The  determination  was  made,  under  the  author's  direction,  by 
Mr.  W.  Eves  and  Mr.  R.  T.  Roche,  in  January,  1912. 

The  object  of  the  measurement  was  to  determine  the  specific 
resistance  of  pure  translucent  celluloid  at  one  temperature. 

The  precision  sought  was  not  very  high  because  it  was  assumed 
that  a  compound  material  of  this  nature,  of  which  the  chemical 
constitution  was  unknown,  is  not  of  such  a  definite  character 
as  to  justify  the  expenditure  of  time  which  would  be  required  to 
obtain  a  highly  accurate  determination.  It  was  decided  that  a 
determination  of  the  specific  resistance  in  which  the  maximum 
error  should  not  exceed  10  per  cent  would  meet  the  requirements. 

The  sample  selected  consisted  of  sheets  of  translucent  celluloid 
furnished  by  the  Eastman  Kodak  Company  and  presumably  of 
the  same  quality  as  that  used  for  Kodak  films.  These  sheets 
were  0.0050  cm  thick.  The  sheets  were  made  up  with  tin  foil 
into  a  tightly  compressed  condenser.  The  tin  foil  was  0.0020  cm 
thick  and  the  effective  capacity  area  of  each  sheet  of  tin  foil  was 
54.1  sq.  cms.  The  over-all  thickness  of  the  condenser  was  0.9  cm. 


ART.  814]     THE  MEASUREMENT   OF  HIGH  RESISTANCE         177 


Consequently  there  were  127  sheets  of  foil  and  128  sheets  of 
celluloid. 

The  method  of  measurement  was  that  of  leakage.  As  the  con- 
denser (the  specific  insulation  of  which  was  to  be  determined)  had 
sufficient/  capacity  itself  for  the  test  it  was  unnecessary  to  use 
another  condenser  in  parallel  with  it  while  the  leak  occurred; 
hence  the  condenser  C/  in  Fig.  812a,  now  represents  the  capacity 
of  the  celluloid  condenser  itself,  while  the  resistance  Rm  repre- 
sents the  resistance  thru  the  celluloid  sheets  of  the  condenser. 

The  celluloid  condenser  was  connected  in  the  circuits  repre- 
sented in  Fig.  814. 


i  f 


'Ga 


FIG.  814. 

Here  C/  is  the  condenser  of  celluloid.  Rm  is  the  internal  resistance 
of  its  sheets  in  parallel.  Ga  is  a  D'Arsonval  galvanometer.  V  is 
a  standard  voltmeter.  Ba  is  a  storage  battery  of  a  few  cells,  and 
r  is  a  slide  rheostat  for  adjusting  the  voltage  applied  to  the  con- 
denser by  moving  the  slider  p.  K  is  a  highly  insulated  key  which 
in  position  1  charges  the  condenser  to  the  potential  indicated  by 
V,  in  the  middle  position  insulates  it,  and  in  position  2  discharges 
it  thru  the  galvanometer. 

The  condenser  was  charged  for  a  known  time  Tf  seconds.  It 
was  then  instantly  discharged  by  throwing  K  from  1  to  2  and 
the  deflection  d\  noted.  The  condenser  was  then  recharged  for 
the  same  charging  time,  and  insulated  for  such  a  time  T  seconds 
that,  on  discharging,  the  deflection  dz  indicated  that  only  about 
37  per  cent  of  the  original  quantity  of  charge  remained.  This 
operation  was  repeated  for  times  of  charge  varying  from  1  to  5 
seconds.  The  values  of  the  insulation  resistance  of  the  condenser 


178 


MEASURING  ELECTRICAL  RESISTANCE         [ART.  814 


were  then  obtained  according  to  the  theory  given  in  par.  812,  the 
formula  used  being 

T 

Rm  =  -  -r  megohms. 


In  order  to  obtain  the  capacity  of  the  celluloid  condenser,  for 
use  in  the  above  formula,  a  standard  mica  condenser  of  known 
capacity  C8  =  0.44  M.F.  was  used.  This  condenser  was  charged 
and  then  immediately  discharged  thru  the  D'  Arson  val  galvanom- 
eter giving  a  deflection  dm.  The  celluloid  condenser  was  next 
charged  to  the  same  potential  for  a  time  Tf  seconds  and  immedi- 
ately discharged  thru  the  same  galvanometer  giving  a  deflection 
dc.  The  capacities  being  in  the  same  ratio  as  the  two  deflections, 

the  capacity  of  the  celluloid  condenser  was  C/  =  -f-  Cs,  which  is 

dm 

the  value  of  the  capacity  used  in  the  above  formula. 
The  data  obtained  are  given  in  the  following  table: 


T' 
seconds 

T 
seconds 

4 

dz 

Cf 
microfarads 

Rm 

megohms 

1 
1 

0 
5 

di-162 

dz=  55 

2.308 

2.01 

2 
2 

0 
12 

rfi-W 

d2=  60 

2.707 

3.85 

3 
3 

0 
..       14 

di  =  216 
dz=  71 

3.078 

4.09 

4 
4 

0 
16 

di  =  236 
d2=  79 

3.362 

4.36 

5 
5 

0 

18 

d1  =  252 

d*=  84 

3.590 

4.57 

Note.  —  Charging  voltage  V  =  0.6  volt. 
Scale  600  mm  from  mirror, 
microcoulombs 


Temp.  =  16.5°  C. 


K 


galv.  deflection  in  mm 


=  8.55  X  10-3  =  galv.  constant. 


A  separate  test  was  made  to  determine  the  proportionality  of  the 
galvanometer  deflections  and  these  were  found  to  be  proportional 
to  the  quantity  of  electricity  discharged  to  better  than  1  per  cent. 

The  deduction  of  the  results  gives  the  relation  which  was  found 
to  exist  between  the  specific  insulation  resistance  of  celluloid  at 
16.5°  C.  and  the  time  that  the  insulation  is  submitted  to  the  im- 
pressed difference  of  potential. 


ART.  814]      THE   MEASUREMENT   OF  HIGH  RESISTANCE         179 

It  appears  that  the  resistance  of  celluloid  is  not  constant  but 
gradually  increases  more  or  less  assymtotically  with  the  time  that 
a  given  E.M.F.  is  applied  to  it,  within  the  limits  of  time,  from  1 
to  5  seconds,  used  in  the  measurement. 

The  data  for  the  resistivity  are  deduced  from  the  dimensions 
of  the  condenser  and  the  above  observations  as  follows: 

R  =  —  p  =  resistance  of  1  sheet  of  celluloid  of  effective  area  S 

o 

and  thickness  I. 

R  I 

-  =  R'  =  —-  p  =  resistance  of   n  sheets  of  celluloid  joined  in 

parallel,  where  Rf  is  the  resistance  measured  and  given  in  megohms 
in  the  table  above.  Hence,  p  =  -r-  R'.  There  were  127  sheets 

of  tin  foil  which  served  as  electrodes  to  126  sheets  of  celluloid. 
S  =  54.1  sq.  cms  and  I  =  0.005  cm.  Taking  the  values  of  R' 
from  the  5th  column  of  the  above  table,  expressed  in  ohms,  for 
1,  2,  3,  4  and  5  seconds  charge  we  obtain,  as  the  specific  resist- 
ance of  celluloid, 

Pi  =  2.69  X  1012  ohms,  p4  =  5.84  X  1012  ohms, 

p2  =  5.06  X  1012  ohms,  p5  =  6.12  X  1012  ohms. 

p3  =  5.48  X  1012  ohms, 

The  specific  resistance  is  seen  to  vary  as  the  time  of  the  applica- 
tion of  the  E.M.F.  in  the  manner  indicated. 

The  reliability  and  precision  of  the  above  results  cannot  be 
estimated  with  any  great  exactness  because  only  one  set  of  meas- 
urements was  made  and  only  one  method  was  used.  In  this  case, 
however,  it  is  very  probable  that  all  the  standards  used  were 
accurate  and  it  is  certain  that  the  observations  were  made  with 
care,  and  the  calculations  were  checked  by  two  computations. 
The  method  is,  however,  open  to  objections  and  both  the  quantity 
measured  and  the  capacity  value  used  in  the  formula  are  in- 
definite. Thus,  since  the  resistance  of  the  dielectric  of  the  con- 
denser was  found  to  vary  with  the  time  of  applied  potential,  Is  one 
to  take  the  time  of  charge  or  the  time  of  charge  plus  the  time 
allowed  for  the  leakage  as  the  true  time  that  the  dielectric  is 
submitted  to  a  potential?  Probably  an  unknown  intermediate 
value  for  the  time  of  applied  potential  would  be  more  correct 
than  either. 

This  determination  has  been  given  here  in  full,  as  it  serves  to 


180  MEASURING  ELECTRICAL  RESISTANCE         [ART.  815 

illustrate  the  method  of  measurement  by  leakage,  to  indicate  the 
procedure  recommended  by  the  author  in  recording  a  physical 
measurement,  and  because  it  serves  to  show  the  difficulties  some- 
times involved  in  obtaining  a  precise  result  in  an  actual  case,  tho 
the  method  and  formula  would  lead  one  to  expect  that  a  high 
precision  is  easily  obtainable. 

815.  High  Resistance  Measured  by  Leakage;  Method  III.  — 
The  following  method  of  leakage  for  measuring  either  short  in- 
tervals of  time  of  electric  contact,*  or  high  resistance  is  a  special 
application,  devised  and  used  by  the  author,  of  Lord  Kelvin's 
null  method  of  mixtures: 

The  method  consists  in  charging  two  condensers  with  potentials 
of  opposite  sign,  then  permitting  one  of  the  condensers  to  leak  for 
a  known  interval  thru  the  high  resistance  to  be  measured,  then 
mixing  the  charge  which  remains  with  the  charge  in  the  other 
condenser  and  discharging  what  still  remains  thru  a  galvanometer. 
The  interval  of  leak  may  be  adjusted  until  a  zero  deflection  of  the 
galvanometer  is  secured.  From  the  data  so  obtained  the  value  of 
the  high  resistance  is  deduced. 

For  explaining  the  practical  carrying  out  of  this  method  reference 
is  made  to  Fig.  815a. 

Here  R  is  the  high  resistance  to  be  measured.  C\  and  C2  are 
two,  preferably  equal,  mica  condensers  of  from  0.2  to  0.5  micro- 
farads each,  ri  and  r2  are  two  resistances,  the  former  being  vari- 
able. They  would  ordinarily  lie  between  1000  and  10,000  ohms. 
S  is  a  rocking  switch  which  when  thrown  to  the  left  makes  contacts 
in  the  mercury  cups  1  and  2,  and  when  thrown  to  the  right 
makes  contacts  in  the  mercury  cups  3  and  4.  The  mercury 
stands  higher  in  cup  4  than  in  cup  3  so  contact  will  occur  in  4 
slightly  earlier  than  in  3.  Ba  is  a  source  of  E.M.F.,  and  Ga 
a  D'Arsonval  or  other  type  of  galvanometer.  K  is  a  highly  insu- 
lated single  contact  key  (not  essential  to  the  method). 

If  we  assume  C\  =  C2  and  R  is  without  appreciable  capacity 
the  measurement  would  then  be  made  as  follows:  Let  r\  equal 
approximately  2.7  r2.  With  K  open  or  closed  throw  S  to  the  left 
giving  Ci  and  C2  charges  which  will  have  the  ratio  2.7  to  1 .  Return 
S  to  the  insulate  position  and  with  K  closed  note  the  time  T 

*  Consult,  "On  the  Duration  of  Electrical  Contact  Between  Impacting 
Spheres,"  by  A.  E.  Keanelly  and  E.  F.  Northrup,  Journal  of  the  Franklin 
Institute,  July,  1911. 


ART.  815]      THE  MEASUREMENT   OF  HIGH  RESISTANCE         181 


during  which  Ci  will  be  losing  its  charge  thru  R.  At  the  end  of 
the  interval  T  throw  switch  S  to  the  right.  This  will  mix  the 
charge  that  remains  in  Ci  with  the  charge  in  (72  and  an  instant 
thereafter  discharge  what  remains  thru  the  galvanometer.  If  the 
interval  T  is  so  chosen  that  the  potential  of  C\  sinks  to  the 
potential  of  C2  the  galvanometer  will  show  no  deflection  when  S 


y/WWWVWA        I — WAA/W 

=Hlil.hl  + 


c2 


Ga 


4  Contacts 
before  3 


FIG.  815a. 

is  thrown  to  the  right.  The  galvanometer  will,  however,  probably 
deflect  either  to  the  right  or  to  the  left  with  the  interval  T  which  is 
first  chosen.  The  amount  and  direction  of  the  deflection  will  give 
information  to  enable  a  value  of  T,  for  a  zero  deflection,  to  be 
chosen  closer  upon  a  second  trial.  Three,  or  at  most  four  trials 
should  be  sufficient  to  find  the  value  of  T  to  give  a  zero  deflection. 
Thus,  the  change  in  the  deflection  will  be  very  nearly  proportional 
to  the  change  in  the  time  of  leak,  or 

d,-d2  =  K  (Ti  -  T9). 
Hence, 

d,  -  0  =  K  (Tl  -  T). 
From  these  two  relations  we  find 


„,  _ 


as  the  true  value  for  the  time  of  leak  which  will  make  the  deflec- 
tion zero.     This  relation  may  be  verified  by  a  third  trial. 


182 


MEASURING  ELECTRICAL  RESISTANCE         [ART.  815 


When  this  adjustment  of  the  time  is  completed  the  value  of  the 
resistance  being  measured  is,  in  megohms,  when  C\  is  in  micro- 
farads, 

T 

Rm=  -  -  -  -•  (2) 

2.303  Cilogio- 
TI 

If  the  resistance  R  has  a  capacity  CR,  as  indicated  by  dotted  lines 
in  Fig.  815a,  it  will  be  necessary  to  determine  this  capacity.  It 
is  not,  however,  necessary  to  determine  its  value  separately  but 
only  in  conjunction  with  the  capacity  C\. 

Let  Cp  =  Ci  +  CR.  With  K  closed  throw  S  to  the  left  and  after 
some  definite  interval  throw  S  quickly  to  the  right.  Repeat, 
adjusting  the  resistances  r\  and  r2  until  there  is  no  deflection  of 
the  galvanometer.  Then,  by  the  theory  of  the  method  of  mix- 
tures, if  r\  and  rj  are  the  values  of  r\  and  r2  which  give  a  balance, 
we  have 


We  may  now  proceed  to  determine  the  resistance  R  as  above  but 
the  value  of  Rm  will  now  be  given  by  the  relation 

Rm    =  -  "  -  77—  (3) 

2.303  C2log 


The  series  of  values  and  the  connections  of  a  variable  standard 
mica  condenser  are  usually  such  that  one  box  of  capacities  may 
be  made  to  serve  for  the  two  condensers  C\  and  C2. 


Middle 


JOo 


-llf-i    i  Middle 

— ll — oa 


u  T: 


FIG.  815b. 


A  usual  subdivision  of  a  one-microfarad  standard  condenser 
and  the  manner  of  using  it  in  a  circuit  like  that  of  Fig.  815a  is 
given  in  I  and  II,  Fig.  815b,  where  the  lettering  of  the  two  dia- 
grams corresponds.  This  last  method  of  measuring  a  high  resist- 
ance by  leakage  has  advantages  over  the  others  described  as 
follows : 

The  method  is  a  zero  method  and  hence  independent  of  the 


ART.  815]      THE   MEASUREMENT   OF  HIGH  RESISTANCE         183 

proportionality  of  a  galvanometer.  A  ballistic  galvanometer  is 
not  required,  as  any  type  of  galvanometer  of  moderate  sensibility 
will  serve  as  the  indicating  instrument.  The  sensibility  of  the 
method  may  be  made  anything  desired  by  simply  increasing  or 
decreasing  the  E.M.F.  of  the  source  Ba  (Fig.  815a).  The  final 
value  of  Rm  is  given  in  terms  of  quantities  all  of  which  can  be  very 

C 

precisely  determined.  Thus  ^  is  given  as  the  ratio  of  two  resist- 
ances, and  likewise  the  ratio  —  can  be  far  more  precisely  determined 

rz 

than  the  ratio  of  two  deflections.  Lastly,  if  the  two  mica  con- 
densers Ci  and  C-2,  are  precisely  alike  and  mounted  in  the  same 
box  any  diminution  of  the  charge  of  one  due  to  its  own  leakage  can 
be  assumed  as  practically  the  same  as  the  diminution  of  the  charge 
of  the  other  due  to  its  own  leakage,  hence  the  insulation  resistance 
of  the  two  condensers  does  not  require  consideration.  But  if 
there  is  a  difference  in  the  insulation  resistance  of  each  it  is  readily 
determined. 

The  method  permits  the  easy  determination  of  the  capacity  Cp 
without  any  change  in  the  circuits.  It  may  be  objected  that  the 
method  requires  several  adjustments  of  the  interval  of  leak  to 
secure  a  balance  and  hence  an  undue  expenditure  of  time.  In 
practice  this  objection  hardly  holds  because  an  exact  balance  is 
not  required  as  the  precise  time  that  would  be  required  for  an 
exact  balance  may  be  obtained  by  a  simple  calculation  from  the 
small  deflection  which  finally  remains  after  two  time-adjustments 
have  been  made. 

The  above  method  was  applied  to  the  measurement  of  the  specific 
resistance  at  room  temperature  of  some  samples  of  red  fiber. 
The  samples  were  in  sheets  0.35  cm  thick.  In  the  middle  and 
upon  opposite  sides  of  a  sheet,  square  pieces  of  tin  foil  (much 
smaller  than  the  sheets)  were  fastened.  In  one  case  the  tin-foil 
sheets  were  fastened  down  with  thin  glue  and  in  the  second  case 
they  were  simply  pressed  down  with  weights.  The  tin-foil  sheets 
were  6.28  cms  X  10  cms.  The  results  were 
pi  =  13.7  X  103  megohms  between  opposite  faces  of  a  centimeter 
cube  when  the  tin  foil  was  glued  down,  and 
p2  =  616  X  103  megohms  when  the  tin  foil  was  merely  pressed  on 

the  fiber.     (Compare  Appendix  IV,  5.) 
To  make  these  determinations  the  following  values  were  used  for 


184  MEASURING  ELECTRICAL  RESISTANCE         [ART.  815 

the  quantities  (their  meaning  being  given  in  Fig.  815a) : 
ri  =  3672  ohms,  rz  =  3000  ohms. 
Ci  =  Cz  =  0.5  microfarad. 

With  the  electrodes  glued  on,  the  time  of  leak  for  a  balance 
within  —  4  scale-divisions  was  8  seconds,  and  with  the  electrodes 
pressed  on,  it  was  300  seconds  for  a  balance  within  +10  scale- 
divisions.  With  corrections  in  the  time  made  for  the  small  de- 
flections obtained,  the  above  values  for  the  resistivity  of  fiber, 
expressed  in  megohms,  were  obtained.  The  ratio  of  about  45  to  1 
between  the  resistivity  obtained  with  electrodes  pressed  upon  the 
fiber  and  electrodes  glued  upon  the  fiber  shows  how  the  apparent 
resistivity  of  a  substance  like  fiber  is  affected  by  apparently  slight 
changes  in  the  conditions  under  which  the  test  is  conducted. 


CHAPTER  IX. 


INSULATION  RESISTANCE  OF  CABLES. 

900.  Introductory  Note.  —  In  the  manufacture  and  installa- 
tion of  long  cables,  especially  those  of  the  submarine  type,  the 
insulation  resistance  per  mile  or  per  kilometer  of  the  cables  is 
tested.     This,  however,  is  only  one  of  several  tests  which  are  made 
upon  such  cables.     The  phenomena  observed  in  testing  the  insula- 
tion resistance  are  complicated  and  not  well  understood  and  a  full 
treatment  of  the  subject  would  not  only  be  too  extended  for  this 
work  but  it  belongs  rather  to  the  subject  of  fault  location  and  the 
study  of  the  electrical  properties  of  dielectrics.     We  shall,  there- 
fore, confine  our  treatment  of  insulation-resistance  measurements 
of  long  cables  to  a  brief  outline  of  the  standard  methods  employed 
and  merely  state  the  phenomena  observed,  requesting  the  reader 
to  refer  to  special  publications  upon  the  subject  for  a  fuller  treat- 
ment, and  for  the  explanations  of  the  phenomena,  which  have 
been  attempted. 

90 1 .  Formula  for  Calculating 
the  Insulation  Resistance  of  a 
Cable.  —  If  the  specific  resist- 
ance p  of  the  insulation  of  a 
cable  is  known,  the  insulation 
resistance  per  kilometer  of  the 
cable  may  be  calculated  as  fol- 
lows: 

In  Fig.  901,  which  represents 
the  cross-section  of  an  insulated 
cable,  let  r\  —  radius  of  metal 
core,  rz  =  radius  of  outside  sur- 
face of  insulation,  r  =  radial  FIG.  901. 
distance  from  center  of  core 

to  any  point  p  within  the  insulation,  dr  =  the  radial  depth  of  an 
infinitely  thin  annulus  of  insulation  and  I  =  the  length  of  the 
cable. 

185 


186  MEASURING  ELECTRICAL  RESISTANCE         [ART.  902 

Then  the  resistance  of  the  annulus  is 


Integrating  between  the  limits  r2  and  ri  we  have 


Expressing  in  common  logarithms  and  combining  the  constants 
we  obtain 

0.3664  p          r, 
R=     — i log™--  (3) 

Example.  —  Assume  p  =  4.5  X  1014.  Take  I  =  1  kilometer  = 
105  cms,  TI  =  0.13  cm  and  rz  =  0.53  cm.  These  constants 
apply,  approximately,  to  a  No.  10  B.  &  g.  wire  insulated  with 
rubber  ^  inch  thick. 

^  0.37  X  4.5  X  1014  0.53 

Then  R  =  -  -  logia  ^^5  =  1.016  X  109  ohms  per 

JL  \J  vJ«  JLo 

kilometer  or,  approximately,  1000  megohms  per  kilometer 

902.  Theorem  upon  the  General  Relation  Between  Capacity 
and  Resistance. 

(1)  From  a  unit  charge  of  electricity  in  a  medium  of  constant 

4  7T 

specific  inductive  capacity  K,  —^  lines  of  electrostatic  force  issue. 

A 

This  is  well  established  but  may  be  easily  proved  as  follows:  If 
two  point  charges  e  and  e'  are  located  at  a  distance  r  from  each 
other  in  space,  the  specific  inductive  capacity  of  which  is  K,  then 
the  force  in  dynes  between  these  charges  is,  by  the  law  of  Coulomb, 

ee' 


F  = 


——  • 
Kr* 


Now  by  definition  the  force  at  any  point  in  space  is  equal  to  the 
rate  of  fall  of  the  potential  in  the  direction  of  the  line  of  force  at 
that  point.  If  we  take  the  potential  to  decrease  as  the  distance 
n  measured  along  the  line  of  force  increases,  we  have 


Or  if  ds  is  an  element  of  surface  taken  normal  to  the  lines  of  force 
at  any  point  in  the  space, 

-^-ds  =  ^-2ds.  (3) 

dn          Kr2  ^  ' 


ART.  902]          INSULATION   RESISTANCE   OF  CABLES 


187 


If  we  imagine  one  of  the  charges  surrounded  by  a  sphere  of  radius 
r  and  we  integrate  over  the  surface  of  this  sphere,  the  area  of 
which  is  4  irr2,  we  have 


(4) 


K 


Let  the  charge  e  =  e'  =  I,  then 


(5) 


Eq.  (5)  states  that  the  total  induction,  namely,  the  total  number  of 
lines  of  force  which  issue  from  a  unit  charge  in  space  of  specific 


4  TT  4 

,  is  -^--    If  the  charge  is  e  then 


~ 
A 


lines  of 


inductive  capacity 

force  issue  from  it. 

(2)  If  eris  the  surface  density  of  electricity  upon  any  surface, 
namely,  the  number  of  electrostatic  units  of  charge  per  unit  area, 
then  for  the  unit  of  area  we  have  the  electric  force  or  induction, 

*•__*=!!?.  (6) 

This  states  that  the   electric  force  or  fall  of  potential  from 

4  7T 

the  surface  in  the  direction  of  the  field  is  -^  times   the   surface 


density  of  the  electricity. 
From  Eq.  (6)  we  obtain 


K 


K  dv 


4?r  dn 


(7) 


(3)  Now  let  Fig.  902  represent  an  element  of  space.  Let  a 
current  of  electricity  enter  the  ele- 
ment at  the  face  dxdy,  and  nor- 
mal to  this  face.  If  Id  is  the 
current  density,  then  the  current 
which  enters  the  face  of  the  ele- 
ment will  be 

di  =  Id  dx  dy.  (8) 

Suppose  the  difference  of  potential 


dn 


FIG.  902. 


between  the  two  ends  of  the  element  is  v  — 
length  of  the  element.     Then, 


i,  and  that  dn  is  the 


di  = 


v  — 


188  MEASURING  ELECTRICAL  RESISTANCE         [ART.  902 

where  dR  is  the  resistance  in  ohms  of  the  element  parallel  to  dn. 

But    -      dR  =  £ryp> 

hence, 

,.      T    ,    ,        v  —  vi  ,    , 
,  •  di  =  Id  dx  dy  =  —:  -  dx  dy, 

or 


an  p  dn  p  dn 


where  -  =K',  the  conductivity  of  the  medium  (assumed  constant). 
P 

T£ 

If  we  compare  Eq.  (7)  with  Eq.  (9)  we  see  that  —  in  the  electro- 

static case  corresponds  with  Kf  in  the  electromagnetic  case. 
Thus  any  formula  for  lines  of  electrostatic  induction  becomes  a 
formula  for  lines  of  current  flow  when  we  make  4x0-  =  Id  and 
change  K  into  K'.  Or  one  can  say,  by  comparing  Eq  (7)  and 
Eq.  (9),  that 

pld=j?<r.  (10) 

Eq.  (10)  leads  to  the  important  conclusion  that,  any  formula  which 
gives  the  electrostatic  induction  thru  a  surface  will  express  the  value 
of  the  current  density  multiplied  by  the  specific  resistance  of  the 
medium  when  the  medium  of  specific  inductive  capacity  K  is  replaced 
by  a  medium  of  specific  resistance  p. 

(4)  As  an  example  of  the  above  take  the  general  case  of  the 
relation  between  electrostatic  capacity  and  ohmic  resistance.  In 
general  the  capacity  of  any  two  electrodes  will  be  the  total  charge 
upon  the  surface  of  one  of  them  divided  by  the  difference  of  poten- 
tial between  them.  Or  if  a  is  the  surface  density  at  any  point  of 
one  electrode  and  ds  is  an  element  of  the  surface  of  one  electrode, 
their  electrostatic  capacity  is 

f  fads 

C8t  -  ii-  (ii) 

v  —  Vi 

Now  multiply  both  sides  of  (11)  by  -^,  and  we  have 

pK 


n          I    I  ~l?  a  ds         I    I  Id  ds 
4TrC8t       J  J  pK  ./  •/ 


pK  v  —  vi  v  — 


ART.  903]          INSULATION   RESISTANCE  OF  CABLES 


189 


the  conductance  from  one  electrode  to  the  other.  In  making 
this  statement  we  conceive  the  medium  of  which  the  specific 
inductive  capacity  is  K  to  be  replaced  by  a  medium  of  which  the 
specific  resistance  is  p,  or  we  conceive  the  same  medium  to  have 

this  specific  resistance. 

ance  between  the  two  electrodes,  then 


If  we  write  G  =  -^ ,  where  R  is  the  resist- 
tt 


j. 
R' 


=  J5,    or    RCst  = 


pK 


(13) 


Eq.  (13)  shows  that  in  every  circuit  where  the  electrostatic  lines 
and  the  lines  of  current  flow  pass  thru  the  same  medium  the 
product  of  the  resistance  and  the  capacity  of  the  circuit  is  constant 

and  equal  to  §—• 
4?r 

A  very  simple  case  will  make  this  physically  clear.  Suppose 
we  have  two  parallel  plates,  each  of  area  A,  and  separated  by  a 
distance  d.  Suppose  that  between  these  plates  there  is  a  slab  of 
dielectric  (much  larger  than  the  plates)  and  that  the  specific 
inductive  capacity  of  this  slab  is  K,  and  its  specific  resistance  is  p. 
Then  the  resistance  between  the  plates  is 


The  capacity  of  the  plates  is 


Cat  = 


AK 


(14) 


(15) 


and  we  have 


pd  AK 
A   4ird 


P_K 

47T* 


Namely,  we  cannot  increase  the  electrostatic 
capacity  of  the  plates  by  increasing  the  area 
without  decreasing  the  resistance  between 
them  by  a  like  amount,  nor  can  we  diminish 
the  distance  between  them,  thereby  increasing 
their  capacity,  without  equally  diminishing 
the  resistance  between  them. 

903.  Application  of  Theorem  to  the  Meas- 
urement of  a  High  Resistance  by  Leakage. 
—  Let  it  be  required  to  measure  the  resist- 


i-c;J 


FIG.  903. 


ance  of  any  condenser,  as  C,  Fig.  903,  by  the  method  of  leakage, 


190  MEASURING  ELECTRICAL  RESISTANCE         [ART.  903 

and  assume  first  that  the  only  capacity  used  is  that  of  the 
condenser  itself.     By  Eq.  (3),  par.  811,  the  time  of  leak  is 

T  =  RC\oge^  (1) 

Now  we  have  shown,  Eq.  (6),  par.  812,  that  for  the  best  results, 
the  capacity,  which  has  been  charged  to  the  potential  Vlt  should 

be  allowed  to  leak  until  this  potential  has  fallen  to  —  •  .     Hence, 

6 

in  this  case, 

T  =  RC  loge  e    or,  as  loge  e  =  1, 

T  =  RC.  (2) 

Since  the  product  RC  will  remain  constant  however  we  change  the 
thickness  or  area  of  the  dielectric,  it  follows  that  the  time  of  leak 

required  for  the  potential  to  fall  to  -  of  its  value  is  the  same  for  a 

€ 

condenser  of  any  form,  size  or  number  of  plates,  provided  the 
quality  of  the  dielectric  remains  the  same.  We  may  express  the 
value  of  this  time  as  follows:  In  Eq.  (2)  C  is  expressed  in  farads 
and  R  in  ohms.  If  we  express  C  in  electrostatic  units,  we  have 

T         RC8t 
9  X  1011  ' 

But  by  Eq.  (13),  par.  902, 


hence, 


An  ordinary  case  would  be  one  in  which  p  =  4.5  X  1014  and 
K  =  5,  which  values  placed  in  Eq.  (3)  give 

T  =  199  seconds  or  about  3J  minutes. 

The  conclusion  to  draw  from  this  result  is,  that  (in  measuring  the 
insulation  resistance  of  a  cable  where  the  capacity  of  the  cable 
is  the  only  capacity  which  leaks)  it  will  require  the  same  time  for 

the  charge  to  leak  away  so  -  of  the  original  charge  remains  what- 

6 

ever  dimensions  are  given  to  the  cable  either  in  length  or  in  cross- 
section.  By  using  a  high  charging  potential  the  charge  will  give 


ART.  904]          INSULATION   RESISTANCE   OF  CABLES  191 

a  sensitive  galvanometer  a  full-scale  deflection,  even  tho  the 
cable  is  very  short.  Hence,  there  will  be  no  gain  in  this  case  in 
using  a  long  length  of  cable  over  a  short  one. 

We  may,  however,  use  an  auxiliary  capacity  Ca  as  indicated 
in  dotted  line  in  Fig.  903.  Then  the  total  capacity  which  leaks 
will  be 

Ct    =   Ca  +  Cst    =   Ca  -j-  T p  > 

the  capacity  in  this  relation  being  expressed  in  electrostatic  units. 
By  Eq.  (3)  we  shall  then  have, 

77  =      RCa  pK  .  . 

9X  1011"r47r9X  1011' 
or  Tt  =  RCmf  10-6  +  PK  0.885  X  10~13  (6) 

where  Cm/  is  the  capacity  of  the  auxiliary  condenser  expressed  in 
microfarads. 

Eq.  (6)  shows  -that,  if  Cm/  is  taken  large  the  time  of  leak  is  in- 
creased, but  as  Cmf  may  be  an  accurate  high-resistance  mica  con- 
denser the  capacity  of  which  can  be  accurately  determined,  while 
the  capacity  of  the  resistance  under  measurement  may  not  be 
determinable  with  exactness,  it  may  be  very  advantageous  for  pre- 
cision to  use  an  auxiliary  condenser.  Furthermore,  it  will  not  in 
this  case  be  necessary  to  submit  the  resistance  under  measurement 
to  the  high  potential  which  might  otherwise  be  required  to  secure 
the  necessary  galvanometer  deflections. 

904.  Insulation  Resistance  of  a  Long  Cable  by  Deflection 
Methods.  —  When  the  insulation  resistance  of  a  long  cable,  such 
as  a  section  of  a  submarine  cable,  is  to  be  tested,  it  is  more  cus- 
tomary to  use  a  direct  deflection  method  than  any  other.  A 
systematic  plan  of  procedure  is  usually  followed,  which  may  be 
described  as  follows: 

The  apparatus  used  generally  consists  of  a  sensitive  galvanom- 
eter of  high  resistance,  a  galvanometer  shunt,  a  short-circuit  key 
for  the  galvanometer,  and  a  Rymer-Jones  key,  also  a  standard 
high  resistance  for  calibrating  the  galvanometer.  The  resistances 
commonly  measured  in  practice  may  range  from  1  to  40,000 
megohms.  The  value  of  the  resistance  is  interpreted  in  terms  of 
galvanometer  deflections. 

In  making  the  test  three  operations  are  performed. 


192 


MEASURING  ELECTRICAL  RESISTANCE         [ART.  904 


1st.  The  negative  pole  of  the  battery  is  joined  to  the  cable 
core,  the  galvanometer  being  in  circuit  after  the  first  rush  of 
current  is  over.  The  positive  pole  of  the  battery  is  put  to  earth. 
The  cable  being  in  a  tank  of  water,  its  outside  sheath  is  earthed. 

2d.  The  cable  is  allowed  to  discharge  itself  for  a  certain  time, 
being  disconnected  from  the  battery.  The  deflections  produced 
by  the  discharge  current,  after  the  first  rush  of  current  is  over, 
are  noted. 

3d.  The  positive  pole  of  the  battery  is  joined  to  the  cable  core 
and  the  negative  pole  to  the  earth,  and  after  the  first  rush  of  cur- 
rent is  over  the  galvanometer  deflections  are  again  noted. 

In  these  three  operations  a  record  of  the  time,  as  well  as  the 
galvanometer  deflections,  is  carefully  kept. 


FIG.  904a. 

For  carrying  out  these  three  operations  the  apparatus  may  be 
connected  as  given  in  Fig.  904a.  K  is  the  Rymer-Jones  key,  and 
it  is  to  be  noted  that  the  levers  of  the  key  cannot  be  placed  in  any 
position  which  will  short-circuit  the  battery.  S  is  the  galvanom- 
eter shunt,  which  may  or  may  not  be  required,  depending  upon 
the  relation  which  the  galvanometer  sensibility  bears  to  the  resist- 
ance being  measured  and  to  the  E.M.F.  of  the  battery  employed. 
k  is  the  short-circuit  key.  The  cable  C  is  usually  coiled  up  in  a 
tank  of  water. 

At  the  point  where  the  wire  from  the  galvanometer  is  attached 
to  the  cable  core,  care  must  be  exercised  to  arrange  matters,  as 
far  as  possible,  so  no  current  will  leak  from  the  core  over  the  out- 


ART.  904]          INSULATION   RESISTANCE  OF  CABLES  193 

side  surface  of  the  insulation  to  the  water  in  the  tank.  A  leakage 
path  of  this  kind  will  be  a  resistance  in  parallel  with  the  resistance 
being  measured  and  its  unrecognized  existence  may  lead  to  en- 
tirely incorrect  results.  Two  methods  are  employed  to  prevent 
this  leak.  The  first  is  to  carefully  pare  the  insulation  at  the  end 
of  the  cable  into  a  conical  shape  and  then,  being  careful  not  to 
touch,  moisten  or  otherwise  contaminate  the  surface,  plunge  the 
end  of  the  cable  into  hot  paraffin  wax.  This  will  give  the  surface 
a  high  insulation.  The  other  method  is  to  employ  a  guard  wire. 
This  wire  is  shown  in  Fig.  904a  in  dotted  line.  Two  or  three  turns 
of  a  fine  wire  are  taken  about  the  conical  end  of  the  insulation  and 
then  carried  to  the  terminal  of  the  galvanometer  which  is  not 
attached  to  the  cable  core. 

When  the  cable  core  is  being  charged  negatively  or  positively 
there  will  be  no  leak  over  the  surface  of  the  insulation  which  will 
pass  thru  the  galvanometer,  because  the  conical  surface  of  the 
insulation  is  maintained  by  the  guard  wire  at  practically  the  same 
potential  as  the  core.  In  the  second  test,  where  the  cable  is 
allowed  to  discharge  itself  for  a  certain  time,  being  disconnected 
from  the  battery,  the  guard  wire  will  only  serve  to  increase  the 
rate  of  leak  and  hence  is  of  doubtful  advantage  for  this  part  of 
the  test. 

It  is  seen  from  the  connections  that  with  the  levers  1  and  2  on  a 
and  6,  respectively,  the  core  of  the  cable  is  put  to  the  positive  pole 
and  the  tank  to  the  negative  pole  of  the  battery.  Putting  levers 
1  and  2  on  b  and  a,  respectively,  puts  the  negative  pole  to  the  core 
and  the  positive  pole  to  the  tank.  Putting  both  levers  on  b 
connects  the  cable  core  and  tank  thru  the  galvanometer  without 
the  battery  being  in  circuit. 

In  the  first  operation,  testing  with  negative  current,  the  short- 
circuit  key  is  kept  closed  while  levers  1  and  2  are  thrown  to  b  and 
a  until  the  first  rush  of  current  is  over,  this  rush  of  current  being 
due  to  the  rapid  charging  of  the  capacity  of  the  cable. 

This  rush  of  current  is  usually  over,  even  in  very  long  cables, 
in  a  few  seconds.  After  about  five  seconds,  the  short-circuit 
key  is  opened.  The  galvanometer  will  now  deflect  to  a  maximum 
deflection,  more  or  less  rapidly,  depending  upon  its  natural  period, 
and  when  it  has  obtained  this  maximum  deflection,  which  will  at 
once  begin  to  decline,  a  record  should  be  made  of  the  deflection 
and  the  time,  counting  the  time  zero  when  the  negative  pole  of 


194  MEASURING   ELECTRICAL  RESISTANCE         [ART.  904 

the  battery  is  put  to  the  core.  The  deflection  of  the  galvanom- 
eter will  steadily  decrease,  rapidly  at  first  and  less  rapidly  as  the 
time  advances,  gradually  reaching  a  minimum  deflection  after 
several  minutes.  It  reaches  its  minimum  or  asymptotic  value  in 
perhaps  30  minutes.  The  test  may  be  continued  with  advantage 
for  this  period,  reading  the  deflections  every  minute. 

The  second  operation  is  now  begun.  The  key  k  is  first  closed 
and  the  levers  of  K  are  thrown  to  6.  A  few  seconds  thereafter  k 
is  opened  again  and  the  deflections  of  the  galvanometer  are  now 
noted  as  in  the  first  operation,  again  counting  the  time  zero  when 
the  levers  of  K  are  changed.  These  deflections  will  be  large  at  first 
and  then  gradually  die  away  as  in  the  first  operation.  This  is 
due  to  the  fact  that  the  electricity,  which  has  been  stored  in  the 
dielectric,  is  given  out,  part  of  it  with  a  rush,  as  a  condenser  dis- 
charges itself,  while,  thereafter,  the  remainder  discharges  slowly 
and  at  a  decreasing  rate. 

This  second  part  of  the  test  may  be  continued  for  five  minutes, 
the  deflections  being  read  every  minute. 

The  third  operation  is  performed  exactly  like  the  first  except 
that  the  positive  pole  of  the  battery  is  put  to  the  cable  core. 
When  the  short-circuit  key  is  opened  it  will  be  noted  that  the 
direction  of  the  deflection  is  opposite  to  that  obtained  when 
charging  the  cable  negatively  and  the  same  as  that  obtained  upon 
discharging  the  cable.  This  part  of  the  test  may  be  continued  for 
five  minutes,  the  deflections  being  recorded  every  minute.  It  is 
called  testing  with  a  positive  current  and  concludes  the  observa- 
tions. The  following  curves,  Fig.  904b,  drawn  from  data  given 
in  "  Kempe's  Handbook  of  Electrical  Testing,"  will  exhibit  the 
general  character  of  the  phenomena. 

Curve  A  exhibits  the  decline  of  the  deflection  with  the  time, 
the  electrification  being  negative,  in  the  interval  1  minute  to  15 
minutes.  The  course  of  the  curve  in  the  interval  0  to  1  minute  is 
not  known.  Beyond  15  minutes  it  may  be  studied  and  is  found 
to  assume  an  asymptotic  value,  as  suggested  by  the  dotted  line. 

Curve  B  exhibits  the  deflections,  where  the  core  and  sheath  of 
the  cable  are  joined,  or  the  cable  is  "  earthed,"  in  the  interval 
1  minute  to  5  minutes.  The  course  of  the  curve  between  0  and 
1  minute  is  not  known.  The  deflection  after  the  fifth  minute  is 
known  to  continue  to  decline  gradually  to  a  zero  value,  as  sug- 
gested by  the  dotted  line.  In  some  instances  it  may  take  many 


ART.  904]          INSULATION   RESISTANCE   OF  CABLES 


195 


hours  for  the  cable  to  become  quite  neutral;  that  is,  for  the 
deflection  to  become  sensibly  zero.  In  a  cable  of  10  or  15  miles 
length  the  deflection  practically  reaches  zero  in  the  course  of  30 
minutes. 

Curve  C  exhibits  the  deflections,  which  gradually  decline  with 
the  time,  when  the  core  of  the  cable  is  put  to  the  positive  pole  of 
the  battery,  immediately  upon  the  removal  of  the  earth  con- 
nection. The  course  of  this  curve  is  like  that  of  curve  A,  and  the 
same  remark  holds,  that  between  0  and  1  minute  the  curve  is  not 
accurately  known,  at  least  not  from  galvanometric  observations. 


FIG.  904b. 

In  cables  without  defects  or  "  faults  "  in  their  insulation  the 
course  of  the  curves  will  be  regular  and  of  the  general  character 
shown  in  Fig.  904b.  Unless  both  ends  of  the  cable  core  are 
joined  to  the  galvanometer,  as  indicated  in  Fig.  904a,  irregularities 
in  the  curves  may  be  produced  by  inductive  effects  which  would 
result,  on  shipboard,  by  the  rolling  of  the  ship,  or,  in  factories,  by 
induction  from  neighboring  currents.  With  both  ends  of  the  core 
joined  to  the  galvanometer  these  effects  are  avoided.  Defective 
insulation  of  the  lead  wires  may  also  produce  irregularities  in  the 
curves,  so  one  must  not  too  hastily  conclude  that  a  cable  is  defec- 
tive when  the  curves  are  seen  to  be  irregular. 


196  MEASURING  ELECTRICAL  RESISTANCE         [ART.  904 

The  quality  of  the  insulation  may  be  judged  from  the  curves 
which  in  a  perfect  cable  exhibit  the  following  relations:  If  the 
current  in  the  earth-reading  (curve  B)  at  the  end  of  the  first 
minute  be  added  to  the,  current  at  the  end  of  the  negative  electri- 
fication (curve  A)  the  sum  should  equal  the  current  for  the  nega- 
tive electrification  at  the  end  of  1  minute.  Or  in  the  case  shown 
by  the  curves,  59  +  142  =  201,  which  is  not  very  different  from 
205. 

Again,  if  the  last  negative  electrification-reading  be  added  to 
the  recorded  earth-reading  at  any  period,  the  sum  should  equal 
the  negative  electrification-reading  at  the  end  of  the  same  period. 
Thus  the  last  electrification-reading  is  142,  the  earth-reading  at 
the  end  of  the  fourth  minute  is  25,  and  the  negative  electrifica- 
tion-reading at  the  end  of  the  fourth  minute  is  164.  We  have 
25  +  142  =  167,  which  is  approximately  the  correct  sum. 

Again,  if  from  the  deflection  at  the  end  of  the  first  minute  of 
positive  electrification  (curve  C)  we  take  the  last  earth-reading, 
it  should  equal  the  deflection  at  the  end  of  the  first  minute  of 
negative  electrification.  Thus,  227  -  22  =  205. 

The  accuracy  of  these  relations,  the  general  smoothness  of  the 
curves,  and  the  rate  at  which  the  deflections  decline  give 
the  necessary  information,  which  enables  those  accustomed  to  the 
requirements  to  judge  of  the  perfection  of  the  insulation  of  the 
cable. 

The  tests  necessary  for  merely  determining  the  specific  resist- 
ance of  the  dielectric  of  the  cable  are  not,  of  course,  as  elaborate 
as  those  given  above,  and  in  ordinary  practice,  when  land  cables 
or  marine  cables  of  moderate  length  are  tested,  only  the  first 
operation  of  testing  with  negative  electrification  is  undertaken. 
The  constant  of  the  galvanometer  and  the  method  of  calculating 
the  specific  resistance  have  already  been  fully  considered.  It 
should  be  noted  here,  however,  that  as  the  resistance  of  the  insula- 
tion apparently,  or  really,  increases  with  the  time  of  electrification, 
one  should  always  give,  in  stating  the  value  of  the  resistance  or 
the  resistivity  of  the  insulation,  the  time  of  electrification  used 
in  obtaining  the  value  given.  It  is  also  found  that  the  resistivity 
is  greatly  affected  by  changes  of  temperature  and  care  should 
therefore  be  used  to  state  the  temperature  which  corresponds  to 
the  value  given.  Thus,  in  stating  the  resistivity  of  gutta  percha, 
we  should  say:  resistivity  =  4.5  X  1014  at  temperature  20°  C. 


ART.  904]          INSULATION   RESISTANCE   OF  CABLES  197 


FIG.  905a. 


FIG.  905b. 


198  MEASURING  ELECTRICAL  RESISTANCE         [ART.  905 

and  was  obtained  by  deflection  method  with  electrification  of 
1  minute. 

905.  Factory  Testing  Set  for  Insulation  Measurements.  — 
Wherever  insulation  measurements  are  to  be  made  frequently,  as 
in  factories  where  cables  are  manufactured,  it  is  necessary,  or  at 
least  very  desirable,  to  provide  a  full  equipment  designed  expressly 
for  the  purpose.  Such  an  equipment,  called  a  "  factory  cable-testing 
set  "  made  by  The  Leeds  and  Northrup  Company  of  Philadel- 
phia, Pa.,  has  been  upon  the  market  for  several  years.  This  set 
is  illustrated  in  Fig.  905a.  Fig.  905b  gives  a  top  view  of  the  set 
which  shows  the  connections. 

The  galvanometer,  and  lamp  and  scale  are  not  shown  in  the 
illustration.  The  chief  features  of  this  set  are:  All  the  instru- 
ments, except  the  galvanometer,  and  lamp  and  scale,  are  mounted 
upon  a  hard  rubber  plate  to  give  high  insulation  between  earth  and 
the  instruments.  The  wire  connections  are  carried  from  the  tops 
of  hard  rubber  posts,  petticoat-insulated.  Except  where  the  wires 
rest  upon  the  posts,  they  are  air-insulated.  The  switches  and 
keys  are  arranged  for  convenient  manipulation. 

The  instruments  of  the  outfit  consist  of  a  D' Arson val  galva- 
nometer of  high  resistance  and  about  1200  megohms  sensibility; 
a  lamp  and  scale,  the  latter  1  meter  long;  a  standard  resistance  of 
0.1  megohm  mounted  in  a  cylindrical  case;  an  Ayrton  universal 
shunt  which  gives  multiplying  powers  of  1,  10,  100,  1000,  10,000 
and  infinity.  There  is  a  highly  insulated  double-pole,  double- 
throw  switch,  and  a  key  which  has  a  lock-down  device  by  which 
it  may  be  permanently  closed.  The  methods  of  using  this  set 
for  insulation  measurements  do  not  differ  in  principle  from  those 
already  described,  while  detailed  instructions  for  its  use  will  be 
furnished  by  the  makers  of  the  set. 


CHAPTER  X. 

RESISTANCE  AS  DETERMINED  WITH  ALTERNATING 

CURRENT. 

1000.  Remarks  upon  Resistance  when  Determined  with  Alter- 
nating Current.  —  We  call  attention  to  the  sense  in  which  the 
term  resistance  is  used  in  this  work.  It  is  a  quantity  which,  in 
general,  may  be  determined  by  direct  currents  and  is  denned  as 
the  fall  of  potential  in  any  portion  of  a  circuit,  divided  by  the 
direct  current  which  flows  in  that  portion  of  the  circuit.  In 
metallic  conductors,  Ohm's  law  is  obeyed,  when  regard  is  had  to 
the  temperature  of  the  conductor. 

When,  however,  the  current  in  a  circuit  is  not  steady,  or  alter- 
nates, there  will  be  a  fall  of  potential  in  any  portion  of  the  circuit 
which  is  the  product  of  a  certain  quantity  Rac  and  that  compo- 
nent of  the  current  which  is  in  phase  with  the  electromotive  force, 
or  more  properly  the  fall  of  potential.  The  quantity  Rac  is  often 
a  different  quantity  than  that  which  is  denned  as  ohmic  resistance. 
It  represents,  in  addition  to  a  true  ohmic  resistance,  anything 
which  causes  energy  losses  of  whatever  character  occuring  in  the 
portion  of  the  circuit  considered.  These  energy  losses  may  be 
due  to  hysteresis  of  iron  in  the  circuit,  to  an  electric  absorption  of 
dielectrics,  to  power  wasted  by  currents  and  electrostatic  poten- 
tials (resulting  in  currents  external  to  the  circuit)  induced  in 
neighboring  circuits,  to  electric  radiation  and  to  other  causes  not 
to  be  classed  with  ohmic  resistance.  In  one  sense  this  quantity 
Rac  is  a  resistance,  but  it  varies  greatly  with  changes  in  fre- 
quency, current  density,  etc.,  and  is  not  to  be  considered  as  a 
constant  quantity  or  as  a  true  ohmic  resistance.  From  a  broad 
viewpoint,  a  work  of  this  character,  which  is  intended  to 
give  a  full  treatment  of  the  methods  of  measuring  resistance, 
should  include,  also,  all  useful  methods  of  measuring  alternat- 
ing-current resistance.  The  subject  is,  however,  so  extensive 
that  we  should  transgress  our  purposes  and  limitations  if  we 
considered  them  here  in  full.  We  shall,  however,  describe  one 

199 


200  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1001 

method,*  devised  by  the  author,  which  will,  in  general,  enable  an 
alternating-current  resistance  to  be  determined  with  sufficient 
precision  to  meet  commercial  requirements.  We  proceed  to  a 
description  of  the  method. 

1001.  To  Measure  an  Alternating-Current  Resistance;  Appa- 
ratus Required.  —  For  carrying  out  this  method  in  a  precise 
manner  the  apparatus  required  is  a  frequency  meter  to  measure 
the  frequency  of  the  current  used  (which  must  be  known,  as  the 
quantity  being  measured  will  vary  with  frequency),  an  alternating- 
current  ammeter  to  give  roughly  the  value  of  the  current  (for  the 
alternating-current  resistance  will  also,  in  general,  depend  upon  the 
value  of  the  current) ,  a  three-point  double-throw  switch  for  quickly 
changing  connections,  resistances,  and  an  electrodynamometer. 
This  last  piece  of  apparatus  should  have  sufficient  capacity  in  its 
current  coils  to  carry  the  full  current  without  heating.  Its  hang- 
ing or  potential  coils  should  be  two  in  number  and  so  arranged  as 
to  form  a  system  which  is  perfectly  astatic  in  respect  to  the  earth's 
field.  The  constant  of  the  instrument  will  then  be  the  same  for 
direct  and  alternating  currents.  All  good  electrodynamometers 
are  constructed  in  this  way.  Either  the  Rowland  deflection  type 
or  Siemens'  type  constructed  to  be  astatic,  may  be  used.  The 
method  to  be  described  was  tested,  using  a  Rowland  deflection- 
type  electrodynamometer.  This  instrument  is  illustrated  in 
Fig.  1001,  and  is  in  part  described  as  follows  in  The  Leeds  and 
Northrup  Company  1911  Catalogue,  No.  74: 

"  The  electrodynamometer  proper  consists  of  two  pairs  of 
fixed  coils  and  a  swinging  coil.  One  pair  of  fixed  coils  on  the  out- 
side of  the  case  is  adapted  to  carrying  currents  as  great  as  50 
amperes;  one  pair  on  the  inside  of  the  case  is  suitable  for  currents 
of  0.1  ampere.  The  swinging  coil  is  adapted  to  currents  as  great 
as  0.1  ampere.  The  position  of  the  fixed  coils  is  such  that  the 
swinging  coil  turns  in  a  field  of  force,  which  is  nearly  uniform 
for  the  angle  thru  which  the  coil  moves.  The  dynamometer  has 
attached  to  it  a  scale  and  telescope  which  are  placed  at  a  fixed 
distance  from  the  mirror  attached  to  the  swinging  coil.  The 
scale  and  telescope  rest  on  an  arm  which  can  be  swung  up  out 
of  the  way  when  the  instrument  is  not  in  use.  The  scale  is  not 

*  This  method  was  first  described  by  the  author  in  a  paper  presented  before 
the  American  Institute  of  Electrical  Engineers  at  the  Boston  Meeting,  June  24- 
28,  1912. 


ART.  1002]     RESISTANCE  WITH  ALTERNATING  CURRENT     201 

provided  with  a  lateral  adjustment.  The  coil  is  brought  to  zero  by 
turning  a  micrometer  screw  which  rotates  the  suspension  tube. 
Thus,  when  the  instrument  is  adjusted  to  read  zero  with  no  current 
flowing,  the  coil  is  always  in  the  same  position  in  the  field.  This 
requirement  must  be  filled  in  order  that  the  constants  of  the  instru- 
ment, after  being  once  determined,  shall  not  alter.  The  hanging- 
coil  system  has  two  coils  connected  to  form  an  astatic  combination 
so  that  the  instrument  will  not  be  affected  in  its  deflection  by  the 
earth's  field  when  used  with  direct  currents." 


FIG.  1001. 


1002.  Description  of  Circuits  and  Theory  of  Method.  —  In 
I  and  II,  Fig.  1002,  GG  are  the  fixed  coils  and  hh  the  hanging, 
astatic  system  of  the  electrodynamometer.  The  hanging  system 
has  an  ohmic  resistance  a  and  there  is  joined  in  series  with  this  a 
non-inductive  resistance  p'.  Let  p  +  a  =  p,  the  entire  resist- 
ance of  the  hanging-coil  system.  In  the  instrument,  illustrated 
in  Fig.  1001,  the  resistance  a  is  about  18  ohms.  It  has  a  minute 
inductance,  which  is  approximately  0.00045  henry.  When  p'  is 
moderately  large  and  non-inductive,  we  may  consider,  without 
sensible  error,  that  the  alternating  current  thru  the  hanging 
system  is  in  phase  with  its  E.M.F.,  even  when  the  frequency  is 
high.  We  shall  so  consider  it  in  all  that  follows. 


202 


MEASURING  ELECTRICAL  RESISTANCE       [ART.  1002 


A  represents  a  coil  which  contains  iron.  It  is  assumed  that 
this  coil  has  a  certain  ohmic  resistance  Rdc  as  measured  by  direct 
current,  and  a  different  resistance  R  as  measured  by  an  alternat- 
ing current  of  a  given  value,  wave  form,  and  frequency.  It  is 
this  latter  resistance  (not  the  impedance  or  inductance  of  A) 
which  the  method  will  enable  us  to  determine.  The  resistance  r 
is  any  resistance  capable  of  carrying  the  full  current.  It  may  be 
a  coil  inductively  wound  but  it  must  not  contain  iron  or  have  such  a 
section  and  resistivity  that  its  resistance  on  alternating  current 
will  be  different  from  its  resistance  on  direct  current  due  to  hyster- 
esis, skin  effect,  or  other  cause.  By  a  sliding  contact  p,  means 
must  be  provided  for  tapping  upon  this  resistance  at  any  point 
along  its  length  as  the  diagram  illustrates,  p  is  a  non-inductive 
resistance,  which  is  equal  to  a  +  p',  the  resistance  of  the  hang- 
ing-coil circuit.  As  will  later  be  shown  the  connections  can  in- 
stantly be  changed  from  the  arrangement  shown  in  I  to  that 
shown  in  II  and  vice  versa. 


FIG.  1002. 


We  wish  first  to  find  the  general  expression  for  the  power 
which  the  wattmeter  measures  when  the  connections  are  those 
shown  in  I,  Fig.  1002.  Call  /  the  current  in  the  fixed  coils  of  the 
dynamometer,  i  the  current  in  the  hanging-coil  circuit,  <£  the  phase 
angle  between  the  currents  i  and  /.  Then  the  deflection  of  the 
dynamometer  is 

(1) 


where  K  is  the  instrumental  constant  of  the  dynamometer.  This 
constant  in  the  case  of  a  deflection  instrument  of  the  Rowland  type 
will  change  slightly  with  the  magnitude  of  the  deflection.  There 
is  also  an  inductive  action  of  the  current  in  the  fixed  coil  which 
tends  to  induce  a  current  in  the  hanging-coil  circuit  when  the 
plane  of  the  movable  system  is  not  vertical  to  the  plane  of  the 
fixed  coils.  This  inductive  action  may  vary  in  a  complicated 


ART.  1002]     RESISTANCE  WITH  ALTERNATING  CURRENT      203 

way,  so  Eq.  (1)  cannot  be  taken  as  strictly  true.  If,  however,  the 
system  is  deflected  by  means  of  the  torsion  head,  when  there  is 
no  current  thru  the  instrument,  so  that  when  current  is  introduced 
the  system  is  brought  back  to  the  position  where  its  plane  is 
vertical  to  the  plane  of  the  fixed  coils,  then  the  inductive  action  is 
null  and  the  relation  given  by  Eq.  (1)  may  be  considered  to  hold 
very  exactly.  In  the  use  of  the  dynamometer  which  follows,  the 
system  should  be  deflected  by  means  of  the  torsion  head,  when 
there  is  no  current  flowing,  to  such  an  extent  that,  on  introducing 
current,  the  instrument  reads  roughly  at  the  zero  of  the  scale. 
With  this  precaution  observed  the  theoretical  relations  will"  be 
found  to  hold  very  exactly. 

If  we  call  V  the  impressed  E.M.F.  between  the  points  a  and 
6,  I,  then  the  current  thru  the  hanging-coil  circuit  will  be 


As  stated  above  the  current  i  will  be  approximately  in  phase  with 
the  E.M.F.,  V,  because  the  inductance  of  the  hanging  coils  is 
very  minute. 

By  Eqs.  (1)  and  (2)  we  have 

D=-IVcos<f>.  (3) 

But  IV  cos  <f>  is  the  entire  power  Wt.  This  power  is  the  sum  of 
two  parts,  W  the  power  consumed  in  A  between  the  points  a  and 
b  and  W  the  power  consumed  in  the  hanging-coil  circuit.  The 
value  of  this  latter  is 

V2 

W  =  — •  (4) 

p 

Thus  we  have 

D=*Wt,  (5) 

D=~p~(W    '  p)' 

From  Eq.  (6) 

and  from  Eq.  (5) 

r,-£a  (8) 


204  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1002 

V2 

—  is  generally  a  small  quantity,     p  is  known  very  precisely  and 

V  can  be  obtained  with  a  voltmeter,  hence  Eq.  (7)  enables  the 
true  power  spent  in  A  to  be  accurately  obtained.  It  is  Eq.  (8), 
however,  which  we  wish  to  use  in  measuring  the  alternate-current 
resistance  of  A. 

With  the  connections  as  shown  in  I,  the  torsion  head  is  turned, 
so  that  with  the  current  (steady  as  possible)  which  is  flowing  the 
deflection  reads  near  the  zero  of  the  scale.  The  total  power  then 
being  registered  is  given  by  Eq.  (8). 

The  connections  are  now  quickly  changed  to  those  shown  in 
II.  The  main  current  will  not  be  altered  by  this  change  in  con- 
nections, for  the  resistance  p  is  simply  made  to  change  places 
with  an  equal  resistance.  The  total  power  which  is  registered, 
however,  will  now  be 

Wt'  =  ^  (9) 

when  D'  is  the  deflection  which  the  dynamometer  now  gives.  The 
contact  p  is  moved  along  the  resistance  r  until  the  deflection 
D'  is  made  equal  to  the  deflection  D,  then  Wtr  will  be  equal  to  Wt. 
Since  the  main  current  I  is  the  same  for  the  connections  I  and  II 
we  have  Wf  =  PR,  =  w^  (10) 

or  R'  =  r'.  (11) 

Here  the  quantity  R'  is  not  the  alternating-current  resistance  of 
the  coil  A  but  it  is  the  alternating-current  resistance  of  this  coil 
when  shunted  with  the  non-inductive  resistance  p.  Similarly  r' 
is  the  alternating-current  resistance  of  r  when  shunted  with  the 
non-inductive  resistance  p. 
We  can  write 

R>  =  J?KK>  and  K  =  -£-*, 

P  +  R  P  +  r 

The  alternating-current  resistance  of  two  parallel  circuits  when 
one  or  both  of  the  branches  contain  reactance  is  not  given  by  the 
same  expression,  as  applies  when  the  branch  circuits  are  without 
reactance,  hence  the  ordinary  expression  for  branch  circuits 

without  reactance,  namely    P.    n  *  must  be  multiplied  by  some 

p  +  K 

factor  Kf,  the  value  of  which  we  now  have  to  determine;  also  the 
factor  k. 


ART.  1002]    RESISTANCE  WITH  ALTERNATING  CURRENT      205 

It  is  shown  in  "  Alternating  Currents  "  by  Bedell  and  Crehore, 
pages  238  to  241,  how  the  alternating-current  resistance,  or,  as 
they  call  it,  the  equivalent  resistance  of  any  number  of  parallel 
circuits  having  self-induction  and  carrying  alternating  current, 
may  be  expressed.  It  is  there  shown  that,  in  general, 

7?'  A 

~A2  +  £V» 

^\      R 


where 

^  _        Ri         |          Rz 

i 

and 

/?„,            Xl         \         X2 

-  -I- 

in  which  expressions  RI,  R%,  etc.,  are  ohmic  resistances  and  xi, 
x2,  etc.,  are  reactances  of  the  several  branches. 

We  can  now  find  expressions  which  will  give  the  values  of  K' 
and  k.     Here  we  have 


and  Bu  = 


We  cannot,  because  of  the  necessity  of  brevity,  give  here  the 
purely  algebraic  processes  required  for  obtaining  the  final  expres- 
sion and  so  we  shall  present  only  the  final  results,  which  are  as 
follows  : 

If/  =  1   j  __  PX* 

' 


Call   the    fractional    expressions    a    and    a\    respectively,    then 
K'  =  1+  a  and  k  =  I  +  ai. 
This  gives,  because  of  Eq.  11, 

R  =       r 

It  will  be  shown  that,  in  general,  when  a  sensitive  electrodyna- 
mometer  is  used,  a  and  a\  are  very  small  quantities  which  in  most 
cases  can  be  neglected. 


206  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1002 

We  have  the  following  cases: 

1st.   a  and  ai  are  negligible.     Then 

R  =  r>  (12) 

2nd.   a  and  on  are  not  negligible  but  are  very  nearly  equal. 
Then,  again, 

R  =  r. 

In  these  two  cases   the   alternating-current  resistance  sought 
may  be  taken  as  numerically  equal  to  the  resistance  r. 
3d.   on  =  0,  but  a  is  not  negligible.     In  this  case 

R  =  r  -  —.  --  (13) 


4th.  a  and  on  are  not  negligible  and  are  unequal,  but  p  is  very 
large.  Then  again  we  can  take  R  =  r. 

Consideration  of  a  single  example  under  case  3d  will  suffice 
to  show  the  magnitude  of  the  error  which  may  be  introduced  by 
omitting  the  correction.  The  example  chosen  is  from  an  actual 
measurement.  With  the  electrodynamometer  available,  only 
0.1  ampere  could  be  passed  through  the  fixed  coil  and  hence,  the 
potential  drop  over  the  coil  A  and  over  the  resistance  r  being  small, 
the  resistance  p  had  necessarily  to  be  taken  very  small  to  give  the 
requisite  sensibility.  If  the  dynamometer  coils  could  have  carried 
several  amperes  (as  is  ordinarily  the  case),  p  would  have  been 
much  larger  and  the  error  would  be  much  less.  In  the  example 
ai  =  0  and 


-[(R  +  P)* 

300  (2  X  3.14  X  60  X  0.036)2 
"  [(11  X  300)2  +  (2  X  3.14  X  60  X  0.036)2]  11  ' 
or  a  —  0.052,  nearly. 
Hence, 

R  =  r  -  l     '       =  0.948  r. 


Thus,  if  we  had  called  R  =  r,  the  error  would  have  been  about 
5.2  per  cent,  R  being  assumed  too  large.  This  conclusion  was 
checked  experimentally.  Without  changing  the  ohmic  resistance 
of  the  coil  A  its  inductance,  which  was  capable  of  variation,  was 


ART.  1003]     RESISTANCE   WITH  ALTERNATING  CURRENT      207 

varied  from  0.003  to  0.036  henry,  and  in  the  first  case,  using  the 
uncorrected  formula,  R  =  10.94  ohms,  and  in  the  second  case, 
using  the  same  formula,  R  =  11.62  ohms,  or  6  per  cent  too  large, 
which  is  in  fairly  close  agreement  with  the  calculated  result  of 
5.2  per  cent. 

If  the  fixed  coils  of  the  dynamometer  had  been  made  to  carry 
10  amperes  instead  of  0.1  ampere,  p  could  have  been  100  times  as 
large,  in  which  case  the  correction  factor  would  reduce  to  about 
0.05  of  1  per  cent. 

The  above  adjustments  having  been  made,  direct  current  can 
be  made  to  replace  the  alternating  current  and  in  the  same  way 
we  find  the  direct-current  resistance  of  A.  It  will  be 

Rdc  =  ri.  (14) 

Hence, 

£-£  <«> 

is  the  ratio  of  the  alternating-current  to  the  direct-current  resist- 
ance of  the  circuit  A.  This  ratio  may  take  a  value  of  2  or  more. 

It  should  be  clearly  understood  just  what  is  meant  by  the 
quantity  R,  which  this  method  measures.  It  is  a  quantity  which, 
expressed  in  ohms  and  multiplied  by  the  square  root  of  the  mean 
square  value  of  the  alternating  current  thru  the  circuit,  expressed 
in  amperes,  will  give  the  square  root  of  the  mean  square  value  of 
that  component  of  the  impressed  E.M.F.  expressed  in  volts,  which 
is  in  phase  with  the  current.  Or  it  is  the  quantity  which,  when 
multiplied  by  the  mean  square  value  of  the  current,  will  give  the 
power  in  watts  which  is  being  dissipated  in  the  circuit.  In  draw- 
ing the  triangle  of  E.M.F.'s  of  an  inductive  circuit  one  sometimes 
represents  the  component  of  the  E.M.F.  which  is  in  phase  with 
the  current  by  the  product  of  the  current  and  the  direct-current 
resistance  RdC.  This  procedure  may  lead  to  considerable  error 
in  circuits  in  which  there  are  other  losses  than  the  I2RdC  losses. 
In  such  circuits  the  alternating-current  resistance  R  should  always 
be  used. 

1003.  Directions  for  Using,  and  Test  of  Method.  —  For  mak- 
ing the  above  measurement  the  apparatus  is  assembled  and  con- 
nected as  shown  in  Fig.  1003. 

D  is  the  electrodynamometer  with  its  hanging  system  h.  The 
heavy  fixed  coils  or  the  light  wire  fixed  coils  are  used  according  to 


208 


MEASURING  ELECTRICAL  RESISTANCE       [ART.  1003 


the  magnitude  of  the  current  which  is  selected  for  the  measure- 
ment. In  the  Rowland  instrument  the  light  wire  fixed  coils  will 
carry  0.1  ampere  and  the  heavy  wire  coils  will  carry  50  amperes. 


(4) 


FIG.  1003. 

C  is  the  alternating-current  ammeter  and  F  the  frequency  meter. 
P  is  a  rheostat  to  control  the  main  current,  s  is  a  switch  to  shift 
from  direct  current  to  alternating  current  and  vice  versa.  S  is 
the  three-point  double-throw  switch  which  in  position  1  makes 
the  connections  shown  in  I,  Fig.  1002  and  which  in  position  2 
makes  the  connections  shown  in  II,  Fig.  1002.  r  is  best  obtained 
from  a  slide-wire  rheostat  of  considerable  current  capacity.  It  does 
not  need  to  be  non-inductive,  but  must  contain  no  iron.  If  its 
reactance  is  just  equal  to  that  of  the  coil  being  measured  a  =  ai 
and  R  =  r  exactly. 

After  the  settings  for  p  have  been  found,  the  connections  are 
broken  at  (3)  and  (4)  and  the  direct-current  resistance  value  of  r 
is  measured  with  a  Wheatstone  bridge  or  by  any  other  convenient 
means. 

The  resistances  p  and  p'  may  be  obtained  best  from  plug  or  dial 
decade  resistance  boxes.  These  may  be  high,  10,000  ohms  or 
so,  depending  entirely  upon  the  current  used,  the  magnitude  of 
the  resistance  being  measured  and  upon  the  sensibility  of  the 
instrument. 

The  torsion  head  may  be  turned  so  that  the  no-current-deflection 
is  between  100  and  200  divisions  of  the  scale.  By  then  adjusting 
P  the  deflection  with  current  on  may  be  made  to  come  near  the 
zero  of  the  scale. 


ART.  1003]     RESISTANCE   WITH  ALTERNATING  CURRENT      209 

It  will  be  found,  if  A  consists  of  an  ironless  variable  standard 
of  inductance,  that  the  variable  standard  may  be  set  to  any 
inductance  value  without  greatly  altering  the  deflection.  The 
change  in  the  deflection  will  be  less  as  p  is  made  larger. 

This  method  will  be  found  useful  in  measuring  the  alternating- 
current  resistance  of  steel-cored  copper  or  aluminum  cables  which 
differ  considerably  from  their  direct-current  resistance. 

The  following  test  was  made  of  this  method. 

This  test  was  made  in  a  manner  to  show  how  large  a  correction 
would  be  required  when  p  is  chosen  only  300  ohms  and  L  is  varied 
between  0.003  and  0.036  henry. 

The  resistance  measured  was  that  of  a  Hartmann  and  Braun 
variable  standard  of  inductance,  which  should,  of  course,  show 
the  same  value  on  direct  and  on  alternating  current  and  at  what- 
ever value  its  inductance  is  set. 

(a)  L  =  0.003  henry. 

With  alternating  current  in  circuit, 

Dt  =  244  (no  current), 
D  =  D'  =  0  (current  flowing), 
R  =  r  =  10.94  ohms, 
7  =  0.08  ampere, 
N  =  60.2  cycles, 
p  =  300  ohms. 

(b)  L'  =  0.036  henry. 

With  alternating  current  in  circuit, 

Dt  =  253  (no  current), 
D  =  D'  =  0  (current  flowing), 
R  =  r  =  11.62  ohms, 
/  =  0.08  ampere, 
N  =  60.2  cycles, 
p  =  300  ohms. 

With  direct  current  in  circuit, 

Dt  =  244  (no  current), 
D  =  D'  —  0  (current  flowing), 
Rdc  =  r  =  10.94  ohms, 
Idc  =  0.08  ampere, 
p  =  300  ohms. 


CHAPTER  XL 

RESISTANCE    MEASUREMENTS    WHEN    THE    RESIST- 
ANCE INCLUDES  AN  ELECTROMOTIVE  FORCE. 

1 100.  Material  Included  Under  this  Title.  —  We  pass  now  to 
a  consideration  of  the  methods  employed  for  the  measurement  of 
resistance,  when  an  electromotive  force  is  included  in  the  portion 
of  the  circuit  the  resistance  of  which  is  to  be  measured. 

The  methods  to  be  considered  are  those  employed  for  measuring 
the  insulation  resistance  of  lighting  or  power  circuits  while  the 
power  is  on,  the  internal  resistance  of  batteries,  the  resistance  of 
the  earth  between  electrodes  where  the  disturbing  effects  of  earth 
currents  must  be  considered,  and  the  resistance  of  electrolytes 
subject  to  an  E.M.F.  of  polarization.  Resistance  measurements 
of  the  above  character  require  special  consideration  at  some 
length. 

noi.  Measurement  of  Insulation  of  an  Electric  Wiring  Sys- 
tem while  the  Power  is  On.  —  It  is  not  infrequently  required 
to  measure  the  insulation  between  the  gas  pipes  and  each  bus-bar 
of  an  interior  wiring  system,  as  that  of  a  city  office  building,  at  a 
time  and  under  circumstances  when  it  is  impracticable  to  shut  off 
the  power. 

Two  methods  are  given  for  making  this  measurement.* 

1102.  Voltmeter  Method  for  Insulation  Measurement  while 
the  Power  is  On.  —  Let  Fig.  1102  represent  any  wiring,  system 
in  which  Xi  and  X2  represent  the  insulation  resistances  between 
the  bus-bars  BI  and  B2)  and  the  earth  (the  gas  or  water  mains 
being  considered  to  have  the  potential  of  the  earth). 

The  diagrams,  I,  II,  III,  of  Fig.  207,  show  circuits  equivalent 
to  the  circuits  represented  by  Fig.  1102.  In  these  diagrams  y 
represents  the  unknown  resistance  of  all  the  lamps,  motors,  etc., 
which  are  connected  across  the  line. 

If  the  bus-bars  are  supplied  with   direct   current    a  Weston 

*  These  methods  were  first  described  by  the  author,  in  the  Electrical  World 
and  Engineer,  May  21,  1904,  vol.  xliii,  pages  966-7. 

210 


ART.  1102]  RESISTANCE  MEASUREMENTS  211 

direct-current  voltmeter  should  be  used.  If  the  current  is  alter- 
nating then  an  alternating-current  voltmeter  of  the  electro- 
dynamometer  type  will  be  required.  The  resistances  Xi  and  Xz 
are  determined  by  knowing  R,  the  resistance  of  the  voltmeter, 
and  by  taking  three  voltmeter  readings  di,  dz  and  D  which  cor- 
respond to  three  voltages  Fi,  Vz,  and  E  indicated  in  Fig.  1102. 
B, 

f 


FIG.  1102. 

1st.  Obtain  the  deflection  D  which  corresponds  to  the  voltage 
E  between  the  bus-bars. 

2d.  Connect  the  voltmeter  between  the  bus-bar  BI  and  the  gas 
or  water  pipe  and  obtain  the  deflection  di  which  corresponds  to 
the  voltage  V\. 

3d.  Connect  the  voltmeter  between  the  bus-bar  B2  and  the 
gas  or  water  pipe  and  obtain  the  deflection  d2  which  corresponds 
to  the  voltage  F2.  The  last  two  readings  should  follow  the  first 
as  quickly  as  possible  so  that  there  will  be  less  chance  of  the  line 
voltage  changing  in  the  intervals  between  the  readings. 

If  the  readings  in  either  of  the  two  latter  cases  are  only  a  frac- 
tion of  a  scale-division,  then  the  insulation  resistance  is  too  high 
to  be  measured  by  this  method  and  one  must  resort  to  the  next 
method  to  be  described.  Having  taken  the  above  three  readings, 

we  obtain  R(D~dl-d,} 

Xl=          ~^~ 
and 

R(D-d1-  dj  f  . 

X2  = ^—  (2) 

The  proof  of  the  relations  (1)  and  (2)  has  been  given  in  par.  207, 
and  a  discussion  is  given  of  the  theoretical  accuracy  obtainable  in 
par.  104. 
•    The  current  I  which  leaks  to  the  ground  will  be 


212  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1103 

In  a  particular  case,  the  insulation  resistance  of  the  wiring  system 
of  a  large  office  building  was  determined.  A  Weston  direct- 
current,  150-  volt  voltmeter  was  used  and  the  following  readings 
and  resistances  were  obtained: 

R  =  12,220  ohms. 

E  =  113  volts  or  D  =  113  scale-divisions. 
Fi  =  1  volt  or  di  =  1  scale-division. 
Vz  =  4  volts  or  dz  —  4  scale-divisions. 
Xi_  12,220  (118  -  1  -4) 

12,220(113-1-4) 


This  example  shows  that  where  the  sum  of  the  resistances  Xi  and 
X  2  is  not  over  a  million  ohms,  the  voltmeter  method  is  sufficiently 
accurate  for  the  purpose  of  deciding  if  insulation  specifications 
have  been  met. 

If  one  side  of  the  line  is  grounded,  that  is,  if  X2  =  0,  we  have, 
V2  =  0,  and  E  =  V\  +  V%  =  V\  and  the  method  fails  to  give  XL 

Any  instrument,  as  a  galvanometer,  in  which  the  deflections 
are  proportional  to  the  current  thru  it  and  which  has  sufficient 
resistance  in  series  with  it  so  that  it  will  at  no  time  deflect  off  its 
scale,  may  be  substituted  for  a  voltmeter. 

1103.  Galvanometer  Method  for  Insulation-Measurement 
while  Power  is  On.  —  This  method  may  be  used  when  greater 
accuracy  is  required  or  when  the  insulation  resistance  to  earth, 
of  at  least  one  side  of  the  line,  is  over  a  megohm. 

The  wiring  system  is  represented  in  I,  Fig.  1103,  and  II,  Fig. 
1103  gives  equivalent  circuits. 

The  method  is  carried  out  by  connecting  across  the  bus-bars  a 
moderately  high  resistance.  A  point  p  is  found  on  this  resist- 
ance where  the  potential  due  to  the  generator  is  the  same  as  that 
of  the  earth.  Then,  with  the  aid  of  a  sensitive  galvanometer  and 
an  external  source  of  E.M.F.,  the  '  resistances  to  earth  r\  and  r2 
are  measured  in  the  following  manner:  k  is  a  key  and  S  an  Ayrton 
universal  shunt.  This  latter  may  be  omitted  if  the  source  of 
E.M.F.  can  be  varied  in  a  known  manner. 

It  is  evident  from  II  that  a  balance  will  be  obtained  when  •=•  =  —  » 

o      rz 

the  key  k  being  in  its  upper  position.     If  k  is  now  depressed,  the 


ART.  1103] 


RESISTANCE   MEASUREMENTS 


213 


resistance  R  encountered  by  the  current  generated  by  the  source 
e  will  be 

R  =  gi+— — —  >  (1) 


b  +  r2  +  a  +  n 

where  gi  is  the  resistance  of  the  galvanometer;  but  in  comparison 
with  TI  and  r2,  a  and  b  can  be  neglected,  also  gi,  then 


R  = 


(2) 


I      C 


II 


By  construction, 


FIG.  1103. 


—  =  T  =  N,  a  known  ratio. 
r2      b 


From  the  last  two  relations  we  deduce 


N 


(3) 

IY 

and 

Taking  d  as  the  deflection  of  the  galvanometer  and  K  as  the 
galvanometer  constant,  the  current  thru  the  galvanometer  is 
e       d  D      eK 

-p  =  -j?i    °r    R  =  -j-*  (5) 

K  should  be  denned  as  the  resistance  in  circuit  with  the  galvanom- 
eter (including  its  own  resistance),  such  that  it  will  give,  with 
one  volt,  a  deflection  of  one  scale-division  at  the  distance  at  which 


214  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1104 

the  scale  is  placed  from  the  mirror  during  the  test,  usually  taken 
as  one  meter. 
Then  we  will  have 


and 

r\  =  -     *—j (7) 

Taking  K  equal  to  108  as  an  average  value  for  an  ordinary  D'Ar- 
sonval  galvanometer  and  e  =  100  volts,  N  =  2,  and  d  =  100  scale- 
divisions,  we  have 

100  X  108  (2  +  1)         _n 

r2  =  -   — 0        nn      -  =  150  X  106  ohms,  or  150  megohms, 
2  X  luU 

i  on  v  i  o^  (*)  i  i  ~\ 
Tl  =  -       x        v          -  =  300  X  106  ohms,  or  300  megohms. 

This  example  shows  that  a  galvanometer  of  very  moderate 
sensibility  will  measure  in  this  way  a  very  high  insulation  resist- 
ance. If,  on  the  other  hand,  the  insulation  is  low,  small  battery 
power  may  be  used  or  the  deflection  of  the  galvanometer  can  be 
cut  down  to  0.1,  0.01,  0.001,  0.0001  by  the  Ayrton  shunt.  The 
only  difficulty  likely  to  be  experienced  in  applying  the  above 
method  is  that,  while  making  the  test,  the  relative  values  of  r\ 
and  rz  keep  changing,  due  to  motors  or  lights  being  thrown  on 
or  off  the  line.  In  this  event  it  is  only  possible  to  obtain  a  sort  of 
average  value  for  the  resistance  to  earth  of  each  side  of  the  line. 

1104.  Determination  of  the  Internal  Resistance  of  Batteries.— 
The  resistance  of  an  electrolytic  cell  or  battery  is  by  no  means  a 
constant  quantity,  even  approximately.  It  will  change  with  the 
temperature,  the  age  of  the  cell,  the  current  which  the  cell  is 
giving  and  with  the  total  ampere-hours  of  current  it  has  yielded. 
It  is  a  quantity  which  varies  greatly  with  the  past  history  of 
the  cell. 

The  determination  of  the  internal  resistance  of  a  cell  on  strictly 
open  circuit  requires  special  methods  and  the  information,  more- 
over, when  obtained  has  little  value  because  of  the  variability  of 
the  quantity  measured.  If  a  cell  is  closed  thru  an  external  resist- 
ance R  and  there  exists  in  the  circuit  an  electromotive  force  E 
the  current  /  which  flows  will  be 


ART.  1105] 


RESISTANCE   MEASUREMENTS 


215 


where  X  is  called  the  internal  resistance  of  the  cell.     If  E'  is  the 
fall  of  potential  over  the  external  resistance  alone,  then 

#'          E 
R 

whence, 

(3) 


R  +  X 
R(E- 


-i, 


(2) 


E' 


We  cannot  here  call  X  anything  more  than  a  quantity  which  must 
be  added  to  R  to  satisfy  Eq.  (1).  It  is  not  an  ohmic  resistance, 
as  it  does  not  obey  Ohm's  law,  for  in  general  X  changes  when  the 
current  changes. 

Nevertheless  the  current  output  of  a  cell,  under  given  circum- 
stances, will  depend  largely  upon  this  quantity,  and  it  is  necessary 
therefore  to  be  able  to  determine  its  value  when  the  cell  is  sub- 
jected to  particular  conditions. 

A  determination  of  the  quantity  X,  to  have  value  and  definite- 
ness,  really  involves  the  making  .of  a  test  of  the  cell  in  respect  to 
several  of  its  characteristics.  These  are  its  open  circuit  E.M.F. 
when  the  cell  is  fresh  and  after  it  has  delivered  a  certain  quantity 
of  electricity,  its  E.M.F.  when  closed  thru  a  given  resistance,  its 
rate  of  polarization,  its  rate  of  recovery  from  polarization,  etc. 
A  description,  therefore,  of  methods  of  measuring  the  internal 
resistance  of  a  battery  should  begin  by  showing  how  a  full  record 
of  the  action  of  a  battery  may  be  obtained.  This  record  is  best 
exhibited  in  the  form  of  curves.  A  procedure  for  obtaining  the 
data  for  such  curves'  in  an  accurate  and  convenient  manner  will 
now  be  explained.  It  is  a  well-known  xr\r. 

method  and  may  be  called  the  condenser 
method  of  testing  batteries. 

1105.  Battery  Tests  by  Condenser 
Method.  —  In  the  diagram,  Fig.  1105a, 
G  is  a  ballistic  galvanometer  or  other  like 
instrument  in  which  the  throw  deflections 
are  proportional  to  the  quantity  of  elec- 
tricity discharged  thru  it.  B  is  a  cell  to 


^^ 

CM 

f  TK 

,B 

-AW/WWWV  — 

[• 

K' 


FIG.  1105a. 


be  tested,  R  a  known  resistance  the  fall 
of  potential  over  which  is  to  be  meas- 
ured, K'  a  key  to  put  this  resistance  in  circuit  with  the  cell  B,  and 
K  is  a  charge  and  discharge  key  for  charging  the  condenser  C  and 
discharging  it  thru  the  galvanometer. 


216  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1105 

The  condenser,  which  should  be  a  mica  condenser  of  about  one 
microfarad,  is  first  charged,  K'  being  open,  by  means  of  a  standard 
cell.  A  Weston  cadmium  standard  cell  is  recommended.  This 
has  an  E.M.F.  of  1.0183  volts  and  a  zero  temperature  coefficient 
between  15°  and  35°  C.  The  condenser  is  then  discharged  thru 
the  galvanometer  and  the  throw  deflection  read  either  with  a 
telescope  and  scale  or  lamp  and  scale. 

The  standard  cell  is  now  replaced  by  the  battery  to  be  tested. 
With  K'  open,  the  condenser  is  charged  and  discharged  as  before. 
Then  Kr  is  closed  and  after  an  interval  of  one  minute  the  condenser 
is  again  charged  and  discharged.  Kf  is  maintained  closed,  prefer- 
ably for  60  minutes,  except  that  at  intervals  of  two  minutes  it  is 
opened  just  long  enough  to  charge  and  discharge  the  condenser, 
at  which  intervals  readings  are  taken  which  give  the  E.M.F.  of  the 
cell  upon  open  circuit.  The  condenser  is  also  charged  and  dis- 
charged, Kf  being  closed,  at  intervals  of  two  minutes  which  alter- 
nate with  the  open-circuit  readings. 

Thus,  at  time  0  open-circuit  reading  is  taken,  Kf  being  open; 
at  end  of  1st  minute  closed-circuit  reading  is  taken,  K'  closed;  at 
end  of  2d  minute  open-circuit  reading  is  taken,  Kr  momentarily 
opened;  at  end  of  3d  minute  closed-circuit  reading  is  taken,  K' 
closed;  at  end  of  4th  minute  open-circuit  reading  is  taken,  K' 
momentarily  opened,  and  so  on  until  at  least  60  minutes  have 
elapsed.  At  the  end  of  60  minutes  Kf  is  opened  permanently  and 
open-circuit  readings  are  taken  at  intervals  of  two  minutes.  In 
this  way  data  are  obtained  for  plotting  the  recovery  curve. 

The  deflection  corresponding  to  the  E.M.F.  of  the  standard  cell 
having  been  obtained,  the  other  E.M.F.'s  can  be  calculated  from 
the  deflections  by  simple  proportion.  Thus,  if  Es  is  the  E.M.F. 
of  the  standard  cell  and  da  the  corresponding  deflection,  then  any 
other  E.M.F.,  Ex,  which  gives  a  deflection  dx  is 

*-;!*• 

The  internal  resistance  of  the  cell  at  any  moment  during  the 
60-minute  test  is  now  obtained  as  follows:  Call  the  E.M.F.  of 
the  open-circuit  reading  at  any  moment  E;  that  of  the  closed- 
circuit  reading  at  the  same  moment  (which  is  the  drop  of  potential 
over  the  resistances  R)  E\.  Then  if  X  designates  the  internal 
resistance  sought,  the  following  relations  hold.  There  is  a  total 


ART.  1105]  RESISTANCE  MEASUREMENTS  217 

E.M.F.,  Ej  and  a  fall  of  potential  over  the  external  resistance  EI. 
Therefore,  there  must  be  a  fall  of  potential  E  —  EI  over  the 
internal  resistance  X;  hence, 

X  iRiiE-EiiE 

v      E  —  EI  D  t  . 

or  X  =    —^  -  R.  (1) 

After  the  polarization  curve  given  by  the  different  E.M.F.'s  called 
E,  and  the  terminal  potential  difference  curve  given  by  the  E.M.F.'s 
called  EI  are  plotted,  then  the  data  for  solving  the  equations 
giving  the  points  for  the  internal  resistance  curves  can  be  read 
directly  from  the  ordinates  of  these  curves.  The  current  flowing 
at  any  time  is  simply 


For  a  full  study  of  a  battery  it  should  be  run  completely  out,  tho 
after  the  first  hour  it  would  be  necessary  only  to  take  readings  at 
intervals  very  much  longer  than  two  minutes.  By  joining  a 
number  of  cells  in  series  (to  always  have  sufficient  E.M.F.)  and 
running  them  completely  down  thru  a  voltameter  the  total 
number  of  coulombs  which  a  cell  is  capable  of  giving  could  be 
computed.  Many  types  of  cells  should  also  be  given  an  age-test 
by  giving  them  a  short  run  and  then  setting  them  aside  for 
several  months  to  test  if  any  destructive  local  action  occurs. 
.»  In  Fig.  1105b  are  reproduced  curves,  taken  by  the  author,  upon 
a  Barrett  silver-chloride  cell,  like  those  supplied  for  portable  test- 
ing batteries.  It  will  be  noted  that  the  internal  resistance  of  this 
type  of  cell  rapidly  falls  as  the  silver  chloride  of  poor  conductivity 
becomes  reduced  to  spongy  silver  of  high  conductivity.  The 
polarization  is  small,  the  current  output  increases  for  the  first 
hour  and  the  recovery  is  rapid,  reaching  1  .150  volts.  These  charac- 
teristics, notwithstanding  the  low  E.M.F.,  have  made  this  type  of 
cell  very  popular  as  a  small  battery  for  testing  purposes. 

Dry  cells  of  standard  size,  as  the  "  Mesco,"  have  become  an 
important  commercial  factor,  and  plans  for  testing  and  rating 
them  require  special  consideration.  However,  the  tests  which 
should  be  made  present  no  problems  in  measurement  which  have 
not  been  fully  discussed  and  the  reader  who  desires  further  infor- 
mation upon  this  subject  is  referred  to  a  report  of  a  committee 
of  the  American  Electrochemical  Society  entitled,  "  Standard 


218 


MEASURING  ELECTRICAL  RESISTANCE       [ART.  1106 


Methods  Recommended  for  Testing  of  Dry  Cells."  This  report 
appeared  in  the  proceedings  of  the  society,  vol.  XXI,  page  275, 
1912,  and  is  signed  by  C.  F.  Burgess,  Chairman;  J.  W.  Brown, 
F.  H.  Loveridge,  C.  H.  Sharp,  Committee  on  Dry  Cell  Tests. 


0.04  4  0.4 


000 


FIG.  1105b. 

1106.  Mance's  Method  of  Measuring  the  Internal  Resistance 
of  a  Battery.  —  The  principle  of  this  method  is  similar  to  the  one 
given  in  par.  404  for  measuring  the  resistance  of  a  galvanometer, 
and,  like  it,  the  method  makes  use  of  the  "  second  property  "  of 
the  Wheatstone  bridge. 

O  0 


W 


FIG.  1106. 


The  connections  to  use  are  shown  in  Fig.  1106,  I  or  II.     With 
galvanometers  of  ordinary  sensibility,  the  current  from  a  battery 


ART.  1106]  RESISTANCE  MEASUREMENTS  219 

placed  in  the  bridge  arm  Ob  would  deflect  the  galvanometer 
violently  off  its  scale.  To  avoid  this  a  resistance  r  is  used  in 
series,  and  a  resistance  s  in  shunt  with  the  galvanometer,  r  and  s 
being  so  chosen  that  at  no  time  the  galvanometer  deflects  off  its 
scale.  It-  is  also  necessary  for  good  results,  if  the  connections  I 
are  used,  to  place  a  resistance  W  which  is  approximately  equal 
to  the  resistance  of  the  slide  wire  ab  in  series  with  the  key  K.  If 
this  resistance  is  not  used  the  wire  becomes  shunted  with  practi- 
cally no  resistance.  The  positions  of  the  galvanometer,  together 
with  the  resistances  r  and  s  and  the  key  K  may  be  interchanged 
as  shown  in  II.  In  this  case  the  resistance  W  is  not  needed. 
The  contact  p  is  moved  to  a  position  such  that  the  galvanometer 
deflection  remains  unaltered  whether  the  key  K  is  open  or  closed. 
When  this  position  is  found  we  have,  if  I  is  the  length  of  the  slide 
wire  and  c  the  distance  of  p  from  a, 


X  =  -=R,  (1) 

where  R  is  the  fixed  resistance  in  the  arm  aO  and  X  the  resistance 
of  the  battery  sought. 

In  applying  this  method  with  a  slide-wire  bridge,  it  should  be 
noted  that,  unless  the  resistance  of  the  slide  wire  ab  is  .made 
very  high  (by  winding  in  a  helix  as  described  in  par.  401),  the 
battery  is  yielding  considerable  current  which,  in  some  types 
of  cells,  would  probably  affect  the  internal  resistance,  making 
it  different  than  it  would  be  if  the  cell  were  yielding  a  less 
current. 

In  Mance's  method,  just  given,  as  well  as  in  Kelvin's  method 
for  measuring  the  resistance  of  a  galvanometer,  greater  precision 
may  be  obtained  by  using  two  equal  extension  coils,  as  shown 
by  n\  and  n2,  Fig.  401a  (§  401).  In  this  case  calling  n  the  value 
of  each  extension  coil,  in  terms  of  equivalent  length  of  bridge  wire, 
we  should  use  the  formula  for  Mance's  method, 

X  =n  +  l~CR.  (2) 

n  +  c 

Also  we  should  use  the  same  formula  (§  404)  for  Kelvin's  method, 
in  which  we  replace  X,  the  resistance  of  the  battery,  by  g,  the 
resistance  of  the  galvanometer. 


220 


MEASURING  ELECTRICAL  RESISTANCE       [ART.  1107 


1 107.  Voltmeter  and  Ammeter  Methods  of  Measuring  the 
Internal  Resistance  of  a  Battery. 

Method  I.  With  K'  open  (Fig.  1107a)  close  K  and  read  E}  the 
open  circuit  E.M.F.  of  the  battery  Ba.  With  K'  closed  read  EI, 
the  drop  of  potential  over  R.  Then  the  current  is 

777  77T 

T  •"  1  •" 

=  ~R='R~TX' 

whence 

v      E-E, 


R. 


(1) 


This  method  assumes,  first,  that  the  voltmeter  takes  so  little  cur- 
rent that,  with  K'  open  and  K  closed,  the  cell  may  be  considered 
to  be  upon  open  circuit,  and  second,  that  the  polarization  of  the 
cell  is  so  trifling  that  when  K'  is  closed  the  E.M.F.  of  the  cell 
remains  unchanged.  Neither  assumption  is  justified  in  the  case 
of  many  types  of  cells  that  polarize  readily  and  is  probably  never 
wholly  justified  with  any  type  of  cell.  However,  for  a  rough 
estimate  of  the  condition  of  dry  batteries,  etc.,  it  is  a  satisfactory 
test.  The  voltmeter  should  have  a  full  scale  reading  of  only  from 
2  to  5  volts  for  accuracy  and  the  resistance  R  should  be  roughly 
equal  to  the  internal  resistance  X  of  the  cell.  The  total  internal 
resistance  of  a  battery  of  cells  joined  in  series  may  be  measured  in 
the  same  way,  a  voltmeter  with  a  scale  reading  higher  being  then 
required. 


-VWVNAM/W 


FIG.  1107a. 


FIG.  1107b. 


Method  II.'  The  measurement  may  be  made  using  a  voltmeter 
and  an  ammeter. 

The  cell  Ba  (Fig.  1107b),  a  resistance  R,  the  ammeter  A  and  a 
key  K  are  joined  in  series.  A  low  reading  voltmeter  V  is  con- 


ART.  1108]  RESISTANCE   MEASUREMENTS  221 

nected  to  the  cell  terminals.     With  K  open,  read  E,  the  open 
circuit  E.M.F.     Close  K  and  read  EI  and  the  current  /.     Then, 


E 

+  R         X 


hence,  x  = 


This  result,  in  which  two  E.M.F.'s  and  a  current  are  measured, 
assumes  also  that  the  voltage  E  is  not  altered  by  polarization  of 
the  cell. 

1108.  A  Word  on  Polarities.  —  In  Fig.  1108  let  a  line  which 
carries  a  direct  current  i  have  introduced  in  it,  in  series,  a  direct- 
current  ammeter  A  and  a  cell  B.  This  is  assumed  to  have  an 
unalterable  E.M.F.,  e,  and  a  zero  internal  resistance.  Also  insert 
an  ohmic  resistance  X.  Connect  a  direct-current  voltmeter  V 
at  the  points  1  and  2  to  measure  the  fall  of  potential  between 
the  points  1  and  2  or  2  and  1.  Let  a  be  the  positive  terminal 
and  b  the  negative  terminal  of  the  voltmeter.  Let  the  positive 
terminal  of  the  cell  be  joined  to  the  resistance  X. 


Line. 


-AAAAAAA/ 


x         -H'- 


JLine 


FIG.  1108. 

The  magnitude  and  direction  of  the  current  i  in  the  line  will 
depend  both  upon  the  E.M.F.,  e,  of  the  cell  B  and  upon  other 
E.M.F. 's  which  are  in  the  rest  of  the  circuit.  Let  the  line  current 
i  when  flowing  in  the  direction  from  2  to  1  be  called  positive, 
and  negative  when  flowing  in  the  opposite  direction.  When  the 
potential  (with  respect  to  the  earth)  at  1  is  greater  than  the 
potential  at  2,  call  the  reading  e\  of  the  voltmeter  positive,  and 
when  the  potential  at  2  is  greater  than  the  potential  at  1,  call 
the  reading  e\  of  the  voltmeter  negative.  First,  assume  that  the 
current  i  flows  from  2  to  1.  Then  the  fall  of  potential  from  3 
to  2  is  such  as  to  tend  to  send  a  current  thru  the  voltmeter  from 
a  to  b  and  the  fall  of  potential  from  3  to  1  is  such  as  to  tend 


222  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1109 

to  send  a  current  thru  the  voltmeter  from  6  to  a.     Hence,  the 
voltmeter  will  read 

ei  =  e  -  iX,  (1) 

from  which  we  deduce 

Z-l'.  (2) 


Second,  assume  that  the  current  i  flows  from  1  to  2.  Then  the 
fall  of  potential  from  3  to  2  is  such  as  to  tend  to  send  a  current 
thru  the  voltmeter  from  a  to  6,  as  before,  but  now  the  potential 
rises  from  3  to  1  and  the  potential  fall  thru  X  will  be  such  as  to 
tend  to  send  a  current  thru  the  voltmeter  from  a  to  b.  Hence, 
the  voltmeter  will  read 

ei  =  e  +  iX.  (3) 

We  shall  know,  however,  that  the  current  in  the  second  case  is 
opposite  to  the  current  in  the  first  case  because  the  terminals 
of  the  ammeter  will  have  to  be  reversed  to  obtain  a  reading. 
Therefore,  if  we  follow  the  convention  of  calling  the  current 
positive  when  flowing  from  2  to  1,  and  negative  when  flowing 
from  1  to  2  (or  against  the  polarity  of  the  cell),  then  we  should 
write,  Eq.  (3), 

e\  =  e  —  iXj  as  in  the  first  case. 

In  Eq.  (1)  if  i  =  0,  e\  =  e  as  it  should,  also  e\  will  remain  positive 
as  long  as  iX  is  less  than  e.  If  iX  becomes  greater  than  e  then  e\ 
will  be  negative,  which  fact  will  be  known  from  the  necessity  of 
changing  the  voltmeter  terminals  in  order  to  obtain  a  reading. 

The  above  principles  must  be  kept  in  mind  when  applying  the 
following  volt  and  ammeter  methods  of  measurement:  We  shall 
adopt  the  convention  that  the  current  in  the  line  is  to  be  regarded 
positive  if  it  has  the  direction  it  would  have  if  the  E.M.F.  in  the 
circuit  being  tested  were  the  only  E.M.F.  acting  and  the  volt- 
meter reading  will  be  regarded  as  positive  if  it  reads  with  its 
positive  terminal  joined  to  the  positive  terminal  of  the  circuit 
which  contains  an  E.M.F.  and  is  under  test. 

1  109.  Voltmeter  and  Ammeter  Method  ;  Principle  of  Polarities 
Illustrated.  —  In  Fig.  1109a,  e  is  a  source  of  E.M.F.  which  has  a 
resistance  represented  by  X.  V  is  a  voltmeter.  E  is  an  auxiliary 
cell,  as  a  storage-battery  cell.  A  is  an  ammeter.  With  the  key 


ART.  1109] 


RESISTANCE   MEASUREMENTS 


223 


K  closed  and  the  polarities  oi  the  two  sources  of  E.M.F.  as  shown, 
we  have 

ei  =  e  —  iX,  (1) 

where  e  is  the  E.M.F.  that  the  voltmeter  reads  when  K  is  open, 
and  61  the  E.M.F.  that  it  reads  when  K  is  closed.  Also  i  is  the 
current  which  the  ammeter  reads,  regard  being  given  to  the  sign 
of  i.  From  Eq.  (1)  we  thus  obtain 


x  = 


(2) 


X              +' 

>-     AAAAAAAAA 

? 

<T>  +  r" 

f  v  V-  

-\A/-     -|^ 

_  ^  ^_ 

FIG.  1109a. 

The  polarity  of  E  is  now  reversed,  as  indicated  in  the  figure  by 
the  dotted  lines,,  and  the  value  of  X  is  then  found  to  be 


e  — 


(3) 


The  first  and  second  values  of  the  resistance  will  probably  not 
agree  on  account  of  polarization  of  the  cell  being  tested.  The 
mean  value,  however,  of  X  and  Xi  should  be  taken  as  representing 
the  most  probable  value  of  the  resistance. 


XB  i 
4l  _  +,  _.     +i 

•AVVVW-J  h-rl  I— 


X 


Bi 

-~||-H 


II 


FIG.  1109b. 


This  method  was  tried  with  the  following  observations  and 
results.  Connections  and  polarities  were  made  first  as  in  I,  and 
second  as  in  II,  Fig.  1109b. 

For  polarities  as  in  I, 

i  =  +  0.1287,     e  =  +  2.16,     el  =  -  2.64. 


224 


MEASURING  ELECTRICAL  RESISTANCE      [ART.  1110 


Hence,  X  =  - 

For  polarities  as  in  II, 

ii=-  0.0635,     e=  +  2.16, 
Hence,  2.16  -  4.50 


-  0.0635 


e2  =  +  4.50. 
=  36.85  ohms. 


The  mean  of  X  and  Xi  is  37.05  ohms. 

In  this  experiment  X  was  a  metallic  resistance  and  BI  a  small 
storage  cell.  The  resistance  X  was  measured  upon  a  bridge,  after 
the  test,  and  found  to  equal  37.25  ohms.  Hence,  the  error  in  the 
measurement  by  method  II  was  a  little  over  one  half  of  1  per 
cent,  a  fair  result,  considering  that  the  instruments  used  were  a 
commercial  voltmeter  and  ammeter. 

i  no.  Total  Resistance  of  a  Network  between  Two  Points 
when  the  Branches  of  the  Network  Contain  Unknown  E.M.F.'s. 
—  This  method  was  devised  by  the  author.  In  Fig.  lllOa  let 


V\AAAAr 


!        fl       ,2/2 


^         ^^*0J 


l  -  1          y5 


2/3       63       2/4 

MM— 


FIG.  lllOa. 

2/i,  2/2,  2/3>  •  •  •  2/n  be  any  combination  of  resistances  joined 
together  in  a  network  in  any  manner  whatever.  Let  ei,  62,  e3, 
.  .  .  en  be  E.M.F.'s  associated  with  the  branches  of  the  network 
having  any  values  and  polarities.  The  problem  presented  is  to 
determine  the  resistance  between  the  points  a  and  b.  The  quan- 
tity R  to  be  measured  should  be  the  same  as  the  quantity  which 
would  be  obtained  if  e\,  e^  es,  etc.,  were  all  zero  and  R  is  defined 

aS  rr 

p  -  ^  fn 

ti  —  y  {*} 

Let  P  be  any  resistance  and  A  an  ammeter  which  will  measure 
/.  Let  &  be  a  switch  or  key  which  will  make  connection  with 
either  the  point  1  or  the  point  2. 


ART.  1110] 


RESISTANCE  MEASUREMENTS 


225 


First  put  k  to  point  1  and  read  the  current  /i  on  the  ammeter 
A  and  the  voltage  Vi  on  the  voltmeter  V.  Then  put  fc  to  2  and 
read  the  current  I \  on  the  ammeter  and  the  voltage  V%  on  the  volt- 
meter. In  the  first  case,  if  we  call  EI  the  resultant  E.M.F.  at 
the  points  a  and  6  of  all  the  E.M.F.'s,  ei,  e2,  e3)  etc.,  then,  as  shown 
in  connection  with  Fig.  1108, 

Vl  =  El-  /!#,  (2) 

and  in  the  second  case 

Vi  =  Ei-  IJt.  (3) 

From  Eqs.  (2)  and  (3) 

Fi-F, 


R  = 


/2-/1 


(4) 


In  applying  this  method  careful  attention  must  be  given  to  the 
convention  of  signs  as  explained  in  par.  1108. 

The  only  assumption  made  here,  which  bears  upon  the  precision 
of  this  method,  is  that  the  resultant  E.M.F.  at  the  points  a  and 
6  is  not  altered  by  polarization  of  the  sources  of  E.M.F.'s,  ei, 
62,  e3,  etc.,  when  the  main  current  is  changed  from  7i  to  1 2- 

In  practice  this  method  gives  good  results  under  certain  cir- 
cumstances that  often  arise.  It  is  adapted  to  the  measurement 
of  the  resistance  between  two  conductors,  as  between  a  gas  and 
water  pipe  main  buried  in  the  earth,  when  the  resistance  path  in 
the  earth  is  subject  to  many  local  and  unknown  E.M.F.'s  which 
correspond  to  the  E.M.F.'s,  ei,  ez,  e3,  etc.,  of  Fig.  lllOa. 


—  Current 


Current 


FIG.  lllOb. 

The  following  trial  of  the  method  was  made  by  the  author: 
Circuits  were  made  up  as  indicated  in  Fig.  lllOb.  EI  was  a 
small  storage-battery  cell  and  E  was  also  a  source  of  E.M.F. 


226 


MEASURING  ELECTRICAL  RESISTANCE       [ART.  1111 


from  a  storage  battery.  P  was  a  resistance  to  vary  the  cur- 
rent in  the  line.  r\  and  r2  were  metallic  resistances.  It  was 
required  to  determine,  by  this  method,  the  resistance  between  the 
points  a  and  b.  The  following  table  exhibits  the  readings  and 
the  results  obtained: 

Here,  V\  and  I\  are  the  voltage  and  current  with  K  on  1,  or 
E  in  circuit,  and  V%  and  1 2  are  the  voltage  and  current  with  K 
on  2,  or  E  out  of  circuit. 


Vi 

v, 

/! 

h 

Ohms 
a  to  b 

Notes 

-7.48 

0.000 

0.9160 

0.0000 

8.17 

EI  made  zero. 

-7.50 

0.100 

0.9734 

0.0553 

8.28 

-7.50 

0.100 

0.9751 

0.0504 

8.22 

P  =0 

P  =5.4  ohms. 

3.87 

0.365 

-0.3660 

0.0623 

8.18 

E  =0.75X  original  E. 

3.86 

0.360 

-0.3650 

0.0623 

8.19 

Ei  =  2x  original  EI. 

4.23 

0.150 

-0.4597 

0.0395 

8.16 

P  =3  ohms. 

4.24 

0.150 

-0.4620 

0.0395 

8.15 

EI  reduced. 

-3.93 

0.150 

0.5375 

0.0394 

8.19 

Polarity  of  E  reversed. 

-3.93 

0.150 

0.5376 

0.0388 

8.18 

8.19 

Mean. 

8.178 

True  value. 

1 1 ii.  Alternating-current   Methods   of   Measuring   the   Re- 
sistance   of    a    Battery.  —  While    several    other    direct-current 
methods  of  measuring  the  internal  resistance  of  a  battery  are 
described  in  the  older  treatises,  they  will  not  be  mentioned  here, 
as  they  are  all  very  much  inferior  to  alternating-current  methods 
and  more  troublesome.     Of  the  alternating-current  methods  of 
value,  two  are  bridge  methods  and  one  an  electrodynamometer 
method,  in  which  the  value  of  the  resistance  is  given  as  equal  to 
a  known  metallic  resistance. 

1 1 12.  Bridge  Method.     Telephone  Detector.  —  This  method 
is  similar  to  the  method  of  Kohlrausch,  described  in  connection 
with  Fig.  1120c,  for  measuring  the  resistance  of  an  electrolyte  (see 
also  par.  1124).     The  arrangement  provided  is  intended  for  the 
measurement  of  the  internal  resistance  of  a  battery  when  this  is 
yielding  a  known  current.     As  the  resistance  of  any  battery, 
and  especially  -one  which  polarizes  readily,  varies  with  the  current 
which  it  gives,  it  is  practically  useless   to   obtain   its   internal 
resistance  if  the  current  to  which  this  resistance  corresponds  is 
not  known. 


ART.  1112] 


RESISTANCE  MEASUREMENTS 


227 


The  arrangement  shown  in  the  diagram  will  enable  the  internal 
resistance  to  be  obtained  and  the  current  which  the  battery  gives 
to  be  measured  or  calculated. 

The  source  of  the  measuring  current,  which  is  rapidly  alternat- 
ing, is  a  small  induction  coil  I  which  is  operated  by  one  or  two 
cells  of  battery  B.  The  bridge  is  an  ordinary  slide-wire  bridge 
which  will  be  found  very  suitable  for  the  measurement.  The 
detector  to  indicate  when  the  bridge  is  balanced  is  a  telephone. 
It  is  convenient  if  this  is  provided  with  a  head  band. 


FIG.  1112. 

Connections  are  made  as  indicated  in  Fig.  1112.  There  is 
shown  in  the  diagram  in  dotted  line  a  condenser  C  in  the  main 
circuit  and  a  condenser  Ci  in  the  telephone  circuit.  The  object  in 
using  these  (which  may  be  cheap  paper  condensers  of  one  micro- 
farad capacity)  is  to  confine  the  direct  current  produced  by  the 
cell  being  measured,  to  the  circuit  c  S%  Si  around  the  bridge. 
If  this  is  done  it  becomes  very  easy  to  calculate  the  current  which 
the  cell  is  giving,  if  its  E.M.F.  has  been  determined  by  means  of 
a  voltmeter.  The  condensers  will  not  interfere  with  the  passage 
of  alternating  current  sufficient  for  the  measurement.  If  the  con- 
densers are  omitted  the  current  from  the  cell  may  still  be  deter- 
mined by  inserting  a  low-reading  ammeter  or  millimeter  at  q. 

The  bridge  is  balanced  for  a  minimum  sound  or  no  sound  in 
the  telephone  by  moving  the  contact  p  upon  the  slide  wire  Si,  S*. 
As  the  scale  of  a  slide-wire  or  meter  bridge  is  usually  divided  into 
1000  divisions,  we  have  for  the  internal  resistance  of  the  cell,  the 
expression  n 


where  a  is  the  reading  from  the  end  of  the  scale  nearest  the  cell 


228  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1112 

and  R  is  the  fixed  resistance  in  the  bridge.     This  latter  should  be 
quite  non-inductive  and  preferably  of  about  the  same  magnitude 

as  the  resistance  of  the  cell.     For  values  of  —  —  --  ,  see  Appen- 

1UUU  —  a 

dix  I,  1.     The  current  from  the  cell   (when  the  condensers  are 
used)  is  now  simply 


where  I  is  the  resistance  of  the  bridge  wire. 

It  may  be  required  to  determine  the  value  of  X  when  the  cell 
is  in  circuit  with  a  lower  resistance  than  the  arms  of  the  bridge. 
The  cell  may  be  made  to  yield  a  larger  current  by  joining  to  its 
terminals  a  known  resistance  r  (shown  in  the  diagram  in  dotted 
line).  If  the  bridge  is  now  balanced  we  have 


r+X~  1000  -a 

V  Rr(1  fA\ 

X  =  -- 


The  current  may  be  measured  with  an  ammeter  or  milliammeter 
inserted  at  q,  or,  if  E  is  measured  with  a  voltmeter  and  the  con- 
densers are  used,  it  may  be  calculated  from  the  formula 


For  the  information  given  by  this  measurement  to  have  the  highest 
value,  the  temperature  of  the  cell,  the  current  which  it  gives,  and 
its  corresponding  resistance  should  all  be  recorded.  The  author 
made  a  careful  test  of  this  method  which  developed  points  of 
interest. 

A  meter  bridge  with  a  slide  wire  of  14  ohms  resistance  was 
selected  for  the  test.  The  alternating  current  was  supplied  by  a 
very  small  induction  coil  about  7  cms  long  and  2.5  cms  in  diam- 
eter. It  was  first  run  by  one  and  later  by  two  small  cells  of  storage 
battery.  Its  vibrator  gave  a  high-pitched  note  but  not  loud. 
The  condensers  used  were  a  1  microfarad  and  a  0.5  microfarad  mica 
condenser,  the  latter  being  in  the  current  circuit  and  the  former  in 
the  telephone  circuit.  The  telephone  was  supplied  with  an  ivory 
plug  attached  to  a  cord  which  could  be  used  with  advantage  to 
stop  up  one  ear  to  keep  out  the  sound  of  the  coil. 


ART.  1112] 


RESISTANCE  MEASUREMENTS 


229 


The  method  was  first  tried,  using  for  the  R  and  the  X  resistances 
two  10-ohm  non-inductive  manganin  coils.  It  was  found  under 
these  circumstances  that  the  sound  in  the  telephone  was  sufficiently 
loud,  and  that  a  point  could  be  found  upon  the  slide  wire  which 
gave  complete  silence.  The  loudness  of  the  sound  was  scarcely 
affected  by  cutting- out  of  circuit  the  two  condensers.  The  setting 
could  be  made  to  within  0.5  of  a  millimeter.  The  bridge  balanced 
at  a  =  501,  showing  that,  as  the  coils  were  equal,  the  wire  was 
practically  of  equal  resistance  either  side  of  its  middle  point. 

A  new  Columbia  dry  cell  was  now  tested,  first  without  a  shunt 
and  later  with  a  shunt  of  1  ohm.  The  resistance  R  was  made 
0.123  ohm.  It  was  found  now  that  it  was  impossible  to  obtain 
silence  in  the  telephone  and  that  it  was  difficult  to  set  the  sliding 
contact  closer  than  2  or  3,  and  sometimes  7  or  8  millimeters.  The 
continuance  of  the  sound  in  the  telephone  was  attributed  to  the 
electrostatic  capacity  of  the  cell,  and  this  was  shown  to  be  the  case 
by  putting  an  equal  number  of  cells,  joined  in  series,  in  the  two 
arms  of  the  bridge  when  a  balance  giving  complete  silence  could  be 
obtained,  as  in  the  case  of  metallic  resistances.  It  should  be 
remembered  that  to  accurately  balance  a  bridge  with  alternating 
current  it  is  necessary  that  the  "  time  constant  "  of  its  adjacent 
arms  shall  be  the  same.  The  smaller  the  cell  and  the  higher  its 
resistance  the  more  accurately  and  easily  can  it  be  measured  by 
this  method.  A  set  of  six  dry  cells,  some  very  old,  were  joined  in 
series  and  with  polarities  mutually  opposed,  and  it  was  found  easy 
to  balance  the  bridge  accurately  because  by  the  arrangement  in 
series  the  resistance  was  increased  more  than  the  capacity.  Thus, 
when  one  has  several  cells  it  is  easier  to  measure  the  resistance 
of  a  number  joined  in  series  than  to  measure  the  resistance  of  one. 
Some  of  the  results  obtained  are  recorded  below: 


Columbia  dry  cell  (standard  size) 


Initial  E.M.F.  E  =  1.47  volts 


#  =  0.123  ohm  (1) 
Settings,  a 
373 
372 

372 

372. 3  Mean 

0.0729  ohm,  calculated  resistance 
of  cell. 


#=0.223  ohm  (2) 
Settings,  a 
239 
238 
238 
237 

235 

237. 2  Mean 


0.0693  ohm,  calculated  resistance 
of  cell. 


230  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1113 

Mean  resistance  obtained  by  (1)  and  (2)  is  X  =  0.0711.  De- 
parture from  mean  is  2.5  per  cent.  The  current  flowing  was 
0.1  ampere.  Same  cell  as  above  shunted  with  1  ohm,  or  r  =  1. 

Settings,  a.  R  =  0.123  ohm. 

377 
370 
380 

375.6  mean.  This  gives  -^—^  =  0.0740  ohm, 

or  X  =  0.0799  ohm.     The  current  was  1.4  amperes. 

An  old  Mesco  dry  cell  was  also  measured.  Its  resistance  was 
found  to  be  about  5  ohms,  but  this  resistance  rose  during  the 
measurement. 

From  these  measurements  it  is  to  be  concluded, 

1st.  That  with  cells  of  moderate  size  a  close  setting  for  a  balance 
with  complete  silence  in  the  telephone  is  impossible. 

2d.  That  the  resistance  of  a  dry  cell  is  an  extremely  variable 
quantity. 

3d.  That  the  method  is  well  adapted  to  metallic  resistances, 
small  cells  of  high  resistance  or  to  a  number  of  cells  in  series, 
but  is  not  accurate  to  more  than  from  3  to  5  per  cent  for  low 
resistance,  single  cells. 

1113.  Bridge  Method;  Electrodynamometer  Detector.  —  This 
method  may  be  applied  in  precisely  the  same  way  as  the  method 
for  measuring  the  resistance  of  an  electrolyte  described  in  con- 
nection with  Fig.  1120d,  except  that  a  condenser  of  considerable 
capacity  should  be  in  circuit  with  the  fixed  coil  of  the  electro- 
dynamometer.  No  current  from  the  battery  can  then  flow  thru 
both  the  fixed  and  movable  coil  of  the  electrodynamometer  and 
so  influence  its  deflection.  If  this  instrument  is  of  the  suspended 
coil  type  (as  designed  by  the  late  Prof.  Henry  A.  Rowland  and 
described  in  par.  1001)  it  will  have  ample  sensibility  when  used 
in  this  way.  Thus  on  a  circuit  of  60  cycles  and  100  volts  the 
current  thru  the  fixed  coil  of  the  dynamometer  will  be  very 
approximately 

i  =  ZirNVC  =  6.28  X  60  X  100  X  C  =  3768  C  amperes, 

where  C  is  the  capacity  in  farads  in  the  circuit.     If  C  is  2.5  X  10"6 
the  current  will  be  0.0942  ampere,  which  is  sufficient.     The  direct 


ART.  1114] 


RESISTANCE   MEASUREMENTS 


231 


current  from  the  cell  is  determined  most  simply  by  measuring  it 
with  an  ammeter  or  milliammeter  in  circuit  with  it. 

1114.  Electrodynamometer  Substitution  Method  (Author's 
Method) .  —  This  method  has  a  much  wider  range  of  usefulness 
than  for  the  particular  measurement  here  described.  Its  applica- 
tion to  the  measurement  of  the  effective  resistance  of  a  circuit 
containing  iron  when  carrying  alternating  current  has  been  already 
described  in  Chapter  X,  and  therefore  its  application  to  the  deter- 
mination of  the  internal  resistance  of  a  battery  may  be  indicated 
very  briefly. 

The  electrodynamometer  should  be  of  the  Rowland  type.  The 
cell  under  test  and  accessory  apparatus  are  connected  as  in  I,  II 
and  III,  Fig.  1114. 


Ill 


FIG.  1114. 


Referring  to  diagram  I  it  will  be  noted  that  the  three-point 
double-throw  switch  S  when  in  position  1,  shown  in  full  line, 
makes  the  connections  indicated  more  simply  in  diagram  II,  and 
when  in  position  2,  shown  in  dotted  line,  makes  the  connections 
indicated  more  simply  in  diagram  III. 


232  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1114 

The  introduction  of  the  condenser  C  in  the  main  circuit  is  to 
prevent  any  direct  current  from  the  battery  passing  thru  the 
fixed  coil  of  the  dynamometer.  The  alternating-current  mains 
of  110  volts  may  be  used  as  the  source  of  E.M.F. 

Now  it  is  evident  by  inspecting  diagrams  II  and  III  that  the 
resistance  of  the  cell  Ba  is  equal  to  r,  provided  the  point  p  has  been 
adjusted  upon  the  resistance  r  until  the  deflection  of  the  electro- 
dynamometer  is  the  same  when,  with  the  switch  S,  the  connec- 
tions are  changed  from  II  to  III  and  vice  versa.  This  equivalance 
of  the  resistance  of  the  cell  Ba  and  the  resistance  r  will  hold 
accurately  provided  the  resistance  p'  is  made  equal  to  the  total 
resistance  p  of  the  hanging  coil  circuit.  The  further  assumption 
must  also  be  made  that  the  electrostatic  capacity  of  the  battery 
is  small.  A  considerable  capacity  reactance  in  the  battery  would 
necessitate  a  small  correction  of  the  same  general  character  as 
that  discussed  in  par.  1002.  It  is  thought,  however,  that  the 
magnitude  of  this  correction  would  in  general  be  so  small  that  it 
could  be  entirely  disregarded.  When  the  measurement  can  be 
made  upon  a  number  of  cells  of  the  same  size  and  kind  at  the  same 
time  the  effect  of  reactance  can  be  reduced  to  any  desired  extent 
by  joining  a  number  of  cells  in  series,  for  the  resistance  measured 
will  increase  directly  with  the  number  of  cells  joined  in  series  while 
the  capacity  will  decrease. 

A  trial  of  this  method  was  made  by  the  author.  It  gave  excel- 
lent results  and  a  brief  description  of  the  test  follows : 

Four  Daniell  cells  were  made  up  in  glass  jars  with  porous  cups. 
These  were  joined  in  series  and  not  opposing.  The  internal  resist- 
ance was  measured,  using  the  method  and  connections  shown  in  Fig. 
1114.  The  electrodynamometer  was  of  the  Rowland  type.  Both 
its  fixed  coils  and  hanging  coil  system  had  a  carrying  capacity  of 
0.1  ampere.  The  source  of  current  was  120  volts  A.C.  and  the 
frequency,  at  the  time  of  the  test,  was  59.9  cycles  per  second. 
The  condenser  C  was  a  mica  condenser  of  1.75  microfarads. 
The  current  flowing  in  the  main  circuit  was  0.077  ampere.  The 
dynamometer  deflection  was  218  divisions.  To  make  this  de- 
flection the  same  with  the  connections  first  as  in  III,  and  then  as 
in  II,  it  was  necessary  to  make  r  =  7.64  ohms;  hence,  the  resist- 
ance of  the  four  cells  in  series  was  7.64  ohms,  making  the  aver- 
age resistance  of  each  cell  1.91  ohms.  This  method  gave  good 
results  without  any  difficulty  arising  and  the  sensibility  was 


ART.  1116]  RESISTANCE  MEASUREMENTS  233 

found  to  be  ample.  At  the  time  of  the  test  the  resistance  thru 
which  the  battery  could  flow  was  300  ohms,  this  being  the  value 
given  to  p.  The  method  gave  in  another  trial  with  these  same 
cells  so  connected  that  their  E.M.F. 's  opposed,  under  which  cir- 
cumstances the  cells  yielded  practically  no  current,  the  value  1.81 
ohms  as  the  mean  resistance  for  each  cell. 

1115.  Galvanometer  Deflection  Methods  for  Obtaining  the 
Resistance  of  a  Battery.  —  Though  the  following  methods  are 
well  known  they  are  not  much  used,  for  the  primary  battery  has 
assumed  a  subordinate  position  as  a  source  of  electric  current. 
However,  for  the  sake  of  completeness  in  the  treatment  of  this  sub- 
ject we  shall  describe  them  briefly. 

Many  types  of  modern  cells,  especially  storage-battery  cells, 
have  an  extremely  low  internal  resistance,  and  in  any  of  the 
methods  for  measuring  this  resistance  it  is  very  advantageous, 
when  one  has  a  number  of  similar  cells,  to  join  as  many  of  them  as 
possible  in  series  opposing  their  E.M.F.'s.  In  this  way  the  resist- 
ances of  the  cells  are  added  in  series,  the  electrostatic  capacity  is 
reduced  and  the  resultant  E.M.F.  is  small,  which  permits  of 
smaller  resistances  in  the  circuits  being  used.  Even  when  there 
is  an  even  number  of  cells,  the  resultant  E.M.F.  is  usually  suf- 
ficient to  furnish  enough  current  for  the  measurement  if  an 
ordinary  D' Arson val  galvanometer  is  the  measuring  instrument. 
It  must  be  remembered  that  the  internal  resistance  measured 
includes  the  connecting  wires  to  the  cells  and  the  contact  re- 
sistance under  the  binding  posts.  These  resistances  must  be 
taken  into  account  and  allowed  for  whenever  great  accuracy  is 
desired. 

1116.  Diminished    Deflection    Method. —  The    battery,    of 
which  the  resistance  X  is  to  be  deter- 
mined (Fig.   1116),  is  joined  in  series 

with  a  resistance  and  a  galvanometer. 
Ordinarily  the  galvanometer  must  be 
shunted  with  a  low-resistance  shunt, 
but  where  a  low-sensibility  galva- 
nometer is  used,  as  a  tangent  gal- 
vanometer, and  the  battery  has  a 
c  o  m  p  a  r  a  t  i  v  e  1  y  high  resistance,  the  FlQ 

shunt  may  be  omitted.  Calling  g  the 
resistance  of  the  galvanometer,  and  s  the  resistance  of  its  shunt, 


234  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1116 

E  the  E.M.F.  of  the  battery,  and  7*1  the  resistance  used,  the  value 
of  the  current  which  flows  is 

t!   --  ~  -  —  =  Kd,.  (1) 


Here  d\  is  the  deflection  of  the  galvanometer  and  K  is  a  constant. 
The  value  of  the  resistance  is  now  changed  from  r\  to  r2  and  the 
current  which  then  flows  is 

t,  =  -  £—  —  Kd*.  (2) 


If  the  galvanometer  is  a  tangent  galvanometer,  then  we  must 
write,  instead  of  Kdi  and  Kdz,  ii  =  K  tan  0i  and  iz  =  K  tan  02, 
where  0i  and  02  are  angular  deflections. 

Eqs.  (1)  and  (2)  make  the  assumption  (never  strictly  true)  that 
the  E.M.F.  of  the  cell  remains  unchanged  when  the  current  is 
changed  from  i\  to  it.  From  the  two  relations  above,  we  easily  derive 

ridi  -  rzd2         gs 

A  =  —  j  --  j  —  --  —  •  (6) 

d}  —  di        g  +  s 

If  r2  is  chosen  so  that  d2  =  -^  ,  Eq.  (3)  becomes 


It  should  be  recalled  that  TI  is  the  resistance  that  gives  the 
deflection  di,  and  r2  is  the  larger  resistance  which  halves  it. 

It  sometimes  happens  that  a  calibrated  galvanometer  is  used  to 
read  the  E.M.F.  of  a  thermocouple.  Now  the  resistance  of  a 
thermocouple  will  change  with  its  depth  of  immersion  in  the  hot 
place,  and  with  the  length  of  the  lead  wires  used.  The  above 
method  could  be  conveniently  employed  to  determine  the  resist- 
ance of  the  thermocouple  circuit  from  binding  post  to  binding 
post  of  the  galvanometer.  In  this  case  the  galvanometer  would 
have  no  shunt  and  its  resistance  g  would  be  known.  Also  the 
resistance  r\  would  be  zero  and  thus,  if  a  resistance  r2  is  inserted 
quickly  in  the  thermocouple  circuit  before  the  temperature  of  the 
hot  junction  has  time  to  change  we  would  find,  by  Eq.  (3), 

X^--rt-g.  (5) 


ART.  1117]  RESISTANCE  MEASUREMENTS 

Or  if  r2  is  so  chosen  as  to  halve  the  deflection  d\, 


235 


X  -  r,  -  g.  (6) 

The  above  measurement  is  made  under  the  best  conditions  when 

/•jo 

r\  H -. — •  is  less  than  X. 


1117.  Kelvin's  Method.  —  The  ob- 
ject of  this  modification  of  the  re- 
duced deflection  method  is  to  maintain 
the  deflection  of  the  galvanometer 
unchanged  and  then  it  makes  no  dif- 
ference what  the  law  of  the  deflection 
of  the  galvanometer  may  be.  With 
the  circuits  arranged  as  in  Fig.  1117 
we  have,  for  the  current  thru  the  gal- 
vanometer, 

Es 


FIG.  1117. 


X  (s  +  g  +  r)  +  s  (g  +  r) 


(1) 


The  value  of  the  shunt  is  now  changed  to  si  and  r  is  changed  to  ri, 
so  the  same  current  as  before  goes  thru  the  galvanometer.    Then, 


(2) 


(3) 


If  in  the  second  case  the  shunt  Si  is  made  infinity  then  Eq.  (3) 

becomes 

-r) 


X  Oi  +  g  +  rO  +  si  (g  + 

From  Eqs.  (1)  and  (2) 

,,  _  ssi  (TI—  r) 

/    '    x        '  -i  +  g) 


X  = 


(4) 


The  method,  like  the  former,  assumes  that  the  E.M.F.  of  the 
battery  remains  unaltered  when  the  current  which  it  delivers  is 
changed. 

For  this  method  to  be  applied  practically,  the  galvanometer 
must  be  very  insensitive  or  shunted,  or  two  cells  of  nearly  equal 
E.M.F.  must  be  joined  in  series  with  their  polarities  opposed. 
If  the  galvanometer  is  shunted,  then  in  place  of  g  we  must  use  the 
shunted  resistance  of  the  galvanometer. 


236  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1118 

1118.  Siemens'  Method.  —  The  arrangement  of  circuits  for 
applying  this  method  is  shown  diagrammatically  in  Fig.  1118. 
Here  the  circuit  a  c  b  is  a  slide  wire  of  uniform  resistance  upon 
which  a  contact  c  may  be  moved.  Ba  is  the  battery  of  which  the 


internal  resistance  is  to  be  determined  and  Ga  is  a  galvanometer 
which  has  its  sensibility  reduced  by  any  means  which  does  not 
include  a  series  resistance.  It  may  be  shunted,  in  which  case  the 
value  assigned  to  its  resistance  must  be  that  of  its  shunted  resist- 
ance. 

Let  X  =  the  internal  resistance  of  Ba,  to  be  found, 

g  =  the  resistance  of  the  galvanometer  (or,  if  shunted,  its 
shunted  resistance), 

p  =  resistance  from  point  c  to  point  6, 

q  =  resistance  from  point  c  to  point  a, 

R  =•  resistance  from  point  c  to  point  o, 

E  =  E.M.F.  of  battery, 

/  =  current  from  battery,  and 

i  =  current  thru  galvanometer  circuit. 
For  brevity,  write  Q  =  q  +  X  and  P  =  p  +  g. 
Then  Q  +  P  =  K,  a  constant. 

By  inspection  of  the  diagram  it  will  be  seen  that  if  c  is  moved  to 
6  the  current  thru  the  galvanometer  will  be  greater  than  if  c  is  at 
some  intermediate  point  between  a  and  b.  Also  if  c  is  moved  to 
a  the  current  thru  the  galvanometer  will  again  be  greater  than  if 
c  is  at  some  intermediate  point  between  a  and  b.  Hence,  gener- 
ally, there  is  some  point  c  between  a  and  6  where  the  current  thru 
the  galvanometer  is  a  minimum.  It  is  this  point,  which,  when 


ART.  1118]  RESISTANCE  MEASUREMENTS  237 

found  by  trial,  will  give  the  value  of  the  resistance  sought.     With 
the  contact  at  any  point  c  on  the  slide  wire,  we  have 


T- 

" 


and 


p  \  %   > 
ER 


or,  as 


/o\ 


Q  =  K  -  P, 


ER  (4) 


"  PK  -  P2  +  KR 
It  is  required  to  move  c  until  i  is  a  minimum  or  until  -  =  ii  is  a 

maximum. 
Thus  we  have 


and  this  expression  is  a  maximum  when 

K  =  2P,  (5) 

or  when  Q  +  P  =  2  P,     or    Q  =  P 

That  is,  when 

2  +  ^  =  P  +  0,     or    X  =  g+p-q.  (6) 

Thus  the  value  of  X  is  found  by  moving  the  contact  c  upon  the 
slide  wire  until  the  deflection  of  the  galvanometer  is  reduced  to  a 
minimum.  As  all  the  resistances  are  known  we  can  calculate 
the  current  which  the  battery  is  yielding,  provided  we  know  its 
E.M.F. 
Thus,  Eq.  (1)  becomes 

/  = E  (p  +  g  +  R) ^ 

7/J  (T,  4-  n  -4-  7?^ 

or  /  =7 


g)(2R+p+g) 

Analysis  shows  (see  Kempe,  "  Handbook  of  Electrical  Testing," 
p.  161)  that  the  measurement  is  made  under  the  best  condi- 
tions when  p  +  q  is  not  less  than  the  greater  of  the  two  quan- 
tities R  +  X  and  R  +  g.  Also  R  should  be  less  than  the  greater 


238 


MEASURING   ELECTRICAL  RESISTANCE       [ART.  1119 


of  the  two  quantities  X  and  g,  and  the  galvanometer  resistance 
should  preferably  not  exceed  X. 

1119.  Resistance  of  Electrolytes.  —  The  resistance  of  electro- 
lytes, as  sulphuric  acid,  salt  solutions,  etc.,  could  be  measured 
with  a  Wheatstone  bridge  in  the  usual  way  if  it  were  not  for  the 
fact  that,  as  soon  as  a  measuring  current  passes  thru  the  electro- 
lyte the  electrodes  polarize  and  an  E.M.F.  is  developed  which 
opposes  the  E.M.F.  which  sends  current  thru  the  electrolyte. 
To  understand  this  clearly  consider  the  diagram,  Fig.  1119. 


FIG.  1119. 

Let  C  be  an  electrolytic  cell  in  one  arm  of  a  Wheatstone  bridge. 
Let  a,  b,  c,  d  be  resistances,  and  i  and  ii  be  currents.  If  the 
resistances  a,  b  and  d  are  always  so  chosen  that  the  bridge  is 
balanced,  we  shall  have  the  potential  drop  Va  from  1  to  2  equal 
the  potential  drop  Vb  from  1  to  4.  Also  the  potential  drop  Vc 
from  2  to  3  will  equal  the  potential  drop  Vd  from  4  to  3.  Or 
ai  =  bii,  and  Vc  =  di\,  whence 


a 


or 


(D 


Now  the  potential  Vc  will  be  equal  to  the  current  i  times  the 
resistance  c  of  the  cell,  less  the  opposing  E.M.F.,  E,  of  polariza- 
tion of  the  cell, 
or  Vc  =  ic  -  E.  (2) 


ART.  1119]  RESISTANCE   MEASUREMENTS  239 

It  may  be  assumed  that  for  a  very  small  current  i  which  has 
flowed  for  a  short  time  t  the  E.M.F.  of  polarization  is  propor- 
tional to  the  quantity  of  electricity  that  has  passed  thru  the  cell, 
or,  what  is  the  same  thing, 

E  =  Kit.  (3) 

Hence, 

Vc  =  ic  -  Kit.  (4) 

Putting  this  value  of  Vc  in  Eq.  (1)  we  obtain 

'c  =  ~-\-Kt.  (5) 

This  last  relation  shows  that  the  resistance  c  of  an  electrolyte, 
which  would  be  measured  by  a  balanced  Wheatstone  bridge,  will 
seem  to  increase  with  the  time  that  the  current  i  is  kept  flowing 
thru  the  electrolyte,  and  that  it  will  always  be  higher  than  the 
true  resistance  of  the  electrolyte.  For  this  reason  the  use  of  the 
Wheatstone  bridge  with  direct  current  is  not  suited  to  the  measure- 
ment of  the  resistance  of  an  electrolyte.  If,  however,  an  alter- 
nating current  be  substituted  for  a  direct  current  and  a  detector, 
responsive  to  alternating  current,  be  substituted  for  the  galvanom- 
eter, the  principle  of  the  Wheatstone  bridge  may  be  used  with 
convenience  and  accuracy.  This  is  because  the  E.M.F.  of  polari- 
zation, produced  by  the  current  in  one  direction  and  which  would 
lead  to  a  balancing  of  the  bridge  giving  too  high  a  value  of  the 
resistance,  will,  upon  the  reversal  of  the  current,  either  be  neutral- 
ized or,  if  not  neutralized,  will  lead  to  a  balancing  of  the  bridge 
giving  too  low  a  value  of  the  resistance.  Thus  the  setting  actu- 
ally obtained  for  a  balance  is  the  same,  whether  polarization  is 
neutralized  or  not,  as  would  be  required  were  there  no  polarization. 

Thus,  to  merely  measure  the  resistance  of  an  electrolyte,  with- 
out attempting  to  determine  its  specific  resistance,  it  is  only 
necessary  to  place  the  electrolyte  in  a  vessel  provided  with  two 
electrodes  of  thin  gold  or  platinum  and  connect  this  vessel  into 
one  arm  of  a  Wheatstone  bridge.  The  other  arms  are  resistances 
which  are  highly  non-inductive.  The  bridge  is  balanced  for  alter- 
nating current.  A  telephone  is  a  suitable  detector  and  a  small  in- 
duction coil  with  a  secondary  winding  furnishes  from  its  secondary 
a  very  suitable  source  of  alternating  current.  The  alternating  cur- 
rent obtained  from  a  small  induction  coil  is  filled  with  harmonics 
and  gives  a  clearer  and  sharper  sound  in  the  telephone,  conse- 


240  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1120 

quently  a  more  accurate  balance,  than  an  alternating  current  from 
ordinary  sources  which  is  approximately  sinusoidal.  It  is  usually 
required,  however,  to  obtain  the  resistivity  as  well  as  the  resist- 
ance of  an  electrolyte,  and  to  determine  this  at  a  very  exactly 
known  temperature.  The  apparatus  and  method  of  working  de- 
vised by  Kohlrausch  are,  with  slight  modifications,  best  adapted  for 
this  determination  and  may  be  described  as  follows: 

1 1 20.  The  Method  of  Kohlrausch  for  Measuring  the  Resis- 
tivity of  an  Electrolyte. 

I.   Direct  determination  of  resistivity,  telephone  employed 
as  detector. 

The  vessel  to  contain  the  electrolyte  must  be  so  shaped  and 
the  electrodes  so  located  that  the  length  and  cross-section  of  the 
electrolyte  measured  can  be  ascertained  with  precision.  For 
this  purpose  nothing  is  better  than  a  cylindrical  glass  tube  of, 
as  nearly  as  possible,  uniform  bore.  This  may,  for  example,  be  a 
glass  tube  20  cms  long  and  1  cm  internal  diameter.  It  may  be 
open  at  both  ends.  In  the  bottom  end  is  fitted  an  electrode  of 
gold  or  platinum  which  completely  fills  and  is  parallel  to  a  right 
cross-section  of  the  tube,  but  may  with  advantage  have  some 
very  smalf  holes  thru  it  to  allow  liquid  to  rise  in  the  tube  when 
this  is  partly  submerged  in  a  vessel  of  liquid.  The  other  elec- 
trode is  of  the  same  material  and  should  be  arranged  to  be  mov- 
able along  the  length  of  the  tube  by  means  of  a  rod  of  metal 
(preferably  gold  plated).  The  rod  or  the  tube  should  have  a  scale 
cut  upon  it  so  that  the  distance  between  the  two  electrodes  may 
be  read  from  the  scale.  Diagrammatically  the  arrangement  would 
be  as  suggested  in  Fig.  1120a.  If  this  tube  is  insulated  in  an  elec- 
trolyte so  that  the  upper  electrode,  but  not  the  upper  end  of  the 
tube,  is  submerged,  the  current  will  only  have  the  one  path,  within 
the  tube,  from  one  electrode  to  the  other. 

The  accurate  calibration  of  this  tube  for  cross-section  and  dis- 
tance between  the  two  electrodes,  for  any  setting  of  the  upper 
electrode,  must  be  made  with  care.  Thus  a  plug  may  be  turned  to 
fit  accurately  into  each  end  of  the  tube  if  this  is  round,  and  these 
plugs  may  have  their  diameters  determined  with  a  micrometer 
caliper.  This  will  give  the  cross-section  where  the  plugs  are  fitted. 
The  distance  between  the  electrodes  may  be  determined  with  a 
cathetometer,  or  the  volume  of  the  tube,  for  a  given  length,  may  be 
determined  by  filling  it  with  mercury  and  weighing  this.  From 


ART.  1120] 


RESISTANCE  MEASUREMENTS 


241 


data  so  obtained  the  mean  cross-section  is  readily  calculated.  It 
is  unusual  to  find  a  tube  which  is  not  more  or  less  conical.  Assum- 
ing that  it  is  conical  and  that  Si  is  the  cross-section  at  one  elec- 
trode, Sz  the  cross-section  at  the  other  electrode,  and  that  I  is  the 
distance  between  the  two  electrodes,  then  it  can  easily  be  shown 


FIG.  1120a. 

that  the  expression  for  the  resistance  from  one  electrode  to  the 
other,  when  the  tube  is  filled  with  a  liquid  of  specific  resistance 
P,  is 


(1) 


I 


is  called  the  resistance  capacity  of  the  tube.     Call  this 
quantity  rc.     Then, 

P-?-  (2) 


To  complete  the  arrangement  the  outer  vessel,  containing  the 
electrolyte,  should  be  provided  with  a  stirrer  for  stirring  the 
electrolyte  to  insure  a  uniform  temperature  throughout.  The 


242  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1120 

temperature  may  then  be  read  with  any  type  of  accurate  thermom- 
eter placed  close  to,  but  outside  of,  the  measuring  tube. 

To  make  a  measurement  the  two  electrode-terminals  are  con- 
nected in  one  arm  of  a  Wheatstone  bridge,  which  is  preferably  of 
the  slide-wire  type.  A  special  form  of  slide-wire  bridge  with  a 
long  slide  wire  wound  spirally  upon  a  marble  cylinder  has  been 
upon  the  market  for  several  years.  A  type  of  bridge  of  this  kind, 
in  which  the  marble  cylinder  is  stationary  and  the  contact  made 
movable,  was  designed  by  the  author  and  is  shown  in  Fig.  1120b. 


FIG.  1120b. 

The  cylinder  upon  which  the  bridge  wire  is  wound  is  15  cms  in 
diameter.  There  are  ten  turns  of  wire,  giving  a  total  length  of 
wire  of  470  cms.  The  contact  point  consists  of  a  minute  piece  of 
hardened  steel  mounted  in  a  short  rod  of  manganin  (the  same 
material  as  the  slide  wire,  so  as  to  avoid  thermal  E.M.F.'s  where 
the  apparatus  is  used  for  other  purposes  with  direct  currents). 
This  steel  piece  slides  upon  the  wire  without  abrading  it.  It  is 
not  visible  in  the  illustration,  being  inside  the  protecting  hood. 
The  hood  revolves  upon  a  threaded  spindle,  the  pitch  of  the  thread 
being  equal  to  the  pitch  of  the  groove  in  the  marble  block  on  which 
the  manganin  wire  is  wound.  The  resistance  of  this  wire  is 
approximately  5  ohms.  The  position  of  the  contact  is  read  by 
means  of  the  vertical  glass  scale  shown  in  the  illustration.  Com- 


ART.  1120]  RESISTANCE  MEASUREMENTS  243 

plete  turns  are  read  upon  the  horizontal  lines  of  the  glass  scale  and 
fractions  of  a  turn  are  read  from  the  scale  upon  the  lower  rim  of 
the  hood.  The  latter  scale  is  divided  into  100  scale  divisions, 
each  of  about  0.6  cm.  These  are  divided  into  halves,  so  that  it  is 
easily  possible  to  estimate  to  thousandths  of  one  complete  revolu- 
tion. The  wire  is  made  very  uniform  in  resistance. 


FIG.  1120c. 

The  connections  to  use  are  diagrammatically  shown  in  Fig. 
1120c.  It  is  generally  better  to  join  the  secondary  of  the  induction 
coil  K  to  the  sliding  contact  p  and  the  point  between  the  cell 
C  and  the  rheostat  r,  and  join  the  telephone  P  to  the  other  two 
points  1,  2  of  the  bridge,  than  vice  versa.  A  balance  can  generally 
be  easily  obtained  with  almost  complete  silence  in  the  telephone. 

The  resistance  R  of  the  electrolyte  is  then  R  =  -r  r.     But  as  the 

scale  of  the  slide  wire  is  divided  into  1000  divisions,  a  +  b  = 
1000  or  b  =  1000  —  a,  whence 


As  the  fraction  -  occurs  frequently  in  bridge  measurements 

1UUU  —  a 

a  table  is  given  for  all  values  of  a  from  1  to  1000  (see  Appendix  1,  1)  . 
It  is  desirable,  for  precision,  to  choose  r  as  nearly  as  possible 
equal  to  the  resistance  of  the  electrolyte  being  measured,  and 
then,  for  a  balance,  p  will  come  near  the  middle  of  the  slide 
wire.  This  will  give  greater  accuracy  to  the  measurement  than 
if  it  came  near  either  end.  The  resistance  r  should  be,  as  far  as 
possible,  non-inductive  and  free  from  electrostatic  capacity.  The 
specific  resistance  at  a  particular  temperature  is  obtained  by 


244 


MEASURING  ELECTRICAL  RESISTANCE       [ART.  1121 


dividing  the  value  of  R,  found  for  a  given  temperature,  by  the 
resistance  capacity  rc  of  the  tube. 

If  one  is  not  in  possession  of  the  special  Kohlrausch  bridge 
illustrated  in  Fig.  1120b,  very  good  results  may  be  obtained  by 
using  an  ordinary  straight  wire  slide-wire  bridge.  For  accuracy  in 
reading  to  not  better  than  0.1  to  0.2  of  1  per  cent  the  circular 
slide  wire  shown  in  Fig.  401b,  may  be  used  with  advantage.  As 
the  scale  connected  with  this  slide  wire  is  laid  off  to  read  the 
resistance  directly  in  per  cent  of  the  standard  resistance  all  cal- 
culation is  avoided  by  its  use. 

II.    Electrodynamometer  employed  as  detector: 
In  pla.ce  of  a  small  induction  coil  for  the  current  source  and  a 
telephone  for  a  detector  one  may  use  alternating  current  and  a 
sensitive  electrodynamometer  as  the  detector.     The  proper  way 
to  connect  this  instrument  in  circuit  is  shown  in  Fig.  1120d. 


Cell. 


i — ^MA/vVW 


FIG.  1120d. 

The  reason  for  placing  the  fixed  coil  G  of  the  electrodynamom- 
eter in  the  main  circuit  is  to  increase  the  sensibility  of  the  appa- 
ratus, which  would  be  very  small  if  the  fixed  and  swinging  coils 
were  joined  in  series  and  the  electrodynamometer  then  connected 
in  the  bridge  in  the  usual  way.  In  this  method  the  alternating 
current  may  be  taken  directly  from  the  mains  and  its  value  reduced 
by  a  suitable  resistance  r.  The  method  is  otherwise  carried  out 
in  the  same  manner  as  when  a  telephone  and  small  induction 
coil  are  used. 

The  type  of  deflection  electrodynamometer  recommended  is  the 
one  described  in  par.  1001. 

1 121.  Determination  of  Relative  Resistivities  of  Electrolytes. 
—  For  this  purpose  the  methods  of  making  the  measure- 
ments are  not  different  from  those  just  given.  A  different 


ART.  1121]  RESISTANCE  MEASUREMENTS  245 

form    of   cell   for  holding   the   electrolyte   is,    however,    to   be 
preferred. 

Suppose  the  containing  cell  to  have  any  shape  and  that  it  is 
filled  with  an  electrolyte  of  known  specific  resistance  pt  at  temper- 
ature t°  C.  If  S  is  the  effective  cross-section  of  the  cell  and  I  its 

length  then  Rt  =  -~  pt  is  the  resistance  of  the  electrolyte  at  tem- 

perature t°  C.     Let  the  resistance  be  accurately  determined  at 
the  temperature  t°  C. 

If  the  cell  is  now  filled  with  an  electrolyte  of  unknown  specific 
resistance  7  and  the  resistance  of  this  electrolyte  is  measured  at 
temperature  ti°  C.,  we  have,  as  before, 


Whence,  taking  the  ratio  of  the  two  resistances  so  obtained,  we 
have 

•  TH-lf  P,  (1) 

Eq.  (1)  gives  the  specific  resistance  of  the  electrolyte  being  meas- 
ured at  the  temperature  £1°  C.,  in  terms  of  the  two  resistance 
measurements  and  the  specific  resistance  pt  of  the  standard 
electrolyte  at  the  temperature  t°  C.  This  last  value  can  be  taken 
from  a  table  of  specific  resistances  of  electrolytes  at  different 
temperatures.  But  if  the  specific  resistance  of  the  electrolyte 
being  measured  is  to  be  compared  at  the  same  temperature 
with  that  of  the  standard  electrolyte  it  is  necessary  to  adopt 
either  of  two  procedures.  One,  is  to  arrange  that  the  temperature 
at  which  R  is  determined  shall  be  the  same  as  the  temperature 
at  which  R'  is  determined;  or  R'  may  be  measured  at  two  temper- 
atures, one  a  little  above  the  temperature  t°  C.  and  one  a  little 
below  this  temperature.  Then  the  resistance  that  the  electrolyte 
would  have  at  the  temperature  at  which  the  standard  electrolyte 
was  measured  can  be  determined  by  a  simple  calculation.  Assum- 
ing that  Rf  has  been  determined  in  either  of  these  ways  we  have, 


The  form  of  cell  which  is  found  very  convenient  for  determinations 
of  the  above  class  is  shown  in  Fig.  1121. 


246 


MEASURING  ELECTRICAL  RESISTANCE       [ART.  1121 


Contact  with  the  platinum  or  gold  electrodes  is  made  perma- 
nently with  mercury  in  bent  glass  tubes  as  shown  in  the  figure, 
and  temporary  lead  wires  dip  into  the  mercury.  During  the 
measurements  the  cell  should  be  suspended  in  a  vessel  contain- 
ing water  which  is  well  stirred  and  of  which  the  temperature  can 
be  accurately  taken. 


Platinum    O44- 
or  Gold  ~ 
Electrode 


FlG.  1121. 

The  resistivities  of  a  saturated  solution  of  sodium  chloride 
at  various  temperatures  are  given  in  Appendix  IV,  4. 

The  methods  described  of  measuring  resistivities  of  electrolytes 
furnish  a  means  of  ascertaining  the  concentration  of  a  solution. 
Tables  have  been  constructed  which  give  the  relation  which  has 
been  found  to  exist  at  some  standard  temperature  between  the 
resistivity  and  the  concentration  of  many  solutions.  Hence,  if 
the  nature  but  not  the  concentration  of  any  solution  is  known, 
for  which  tables  exist,  the  latter  can  be  simply  and  accurately 
found  by  measuring  the  resistivity  of  the  solution. 

The  tables  usually  give  not  the  resistivity  but  the  conductivity 
which  is  the  reciprocal  of  the  resistivity.  The  standard  temper- 
ature for  which  the  conductivities  are  given  in  most  tables  is 
18°  C.,  and  hence  measurements  should  be  made  as  near  this 
temperature  as  practicable. 

For  tables  of  the  electrical  conductivity  of  solutions,  the  reader 
is  referred  to  "  Physical  and  Chemical  Constants,"  by  G.  W.  C. 
Kaye  and  T.  H.  Laby,  pages  86-87.  Also  to  "  Physikalisch- 
Chemische  Tabellen,"  by  Landolt,  Bornstein  and  Meyerhoffer, 
page  735  and  following,  third  edition. 

It  may  be  remarked  here  that  the  resistivity  of  a  saturated 
solution  of  sodium  chloride  (NaCl)  at  20°  C.  is  4.4248  ohms. 
This  is  the  resistance  between  opposite  faces  of  a  centimeter- 


ART.  1122]  RESISTANCE   MEASUREMENTS  247 

cube  of  the  solution.  The  resistivity  of  100  per  cent  conductivity 
copper  at  the  same  temperature  is  1.7215  X  10"6  ohm.  Hence 
the  salt  solution  has  a  resistivity  which  is  2.570  X  106  times  that 
of  copper.  In  general,  electrolytes  have,  roughly,  a  million  times 
the  resistivity  of  metals. 

1 122.  Bering's  Liquid  Potentiometer  Method  for  Determining 
Electrolytic  Resistances.  —  Dr.  Carl  Hering  has  shown*  that 
the  principle  of  the  potentiometer  may  be  employed  for  deter- 
mining the  resistance,  and,  if  the  necessary  dimensions  of  the  con- 
taining vessel  are  known,  the  resistivity  of  an  electrolyte.  The 
method  is  said  to  avoid  the  errors  which  in  direct-current  methods 
are  ordinarily  introduced  by  polarization  of  electrodes.  This 
method  permits  the  resistance  to  be  measured  between  two  se- 
lected points  of  a  quantity  of  electrolyte  contained  in  a  tank.  It 
is  applied  as  follows: 

The  apparatus  employed  consists  of  a  tank  of  rectangular  form 
built  of  insulating  material.  A  suitable  scale  is  fastened  to  the 
upper  and  longer  edge  of  the  tank  for  measuring  the  distance 
between  the  potential  electrodes  at  the  moment  a  balance  is 
obtained.  Two  current  electrodes  are  fitted  into  each  end  of  the 
tank  which  reach  across  it,  and  above  the  surface  of  the  liquid. 
The  liquid  may  fill  the  tank  to  any  convenient  height. 

It  is  important  to  choose  potential  electrodes  which  are  as  inert 
as  possible  in  the  electrolyte.  Two  gold  coins  or  thin  strips  of  gold 
are  recommended  for  these  electrodes.  The  instruments  required 
are  an  ammeter,  a  galvanometer  or  millivoltmeter,  and  means  for 
determining  the  E.M.F.  of  the  small  cell  which  is  used  as  the 
standard  of  E.M.F.  Porous  diaphragms  should  be  fitted  in  the 
ends  of  the  tank  to  prevent  the  products  of  decomposition  at 
the  current  electrodes  from  entering  the  main  body  of  the  elec- 
trolyte. The  polarities  of  the  two  batteries  are  arranged  as  in 
the  ordinary  use  of  the  potentiometer.  The  "  setting  "  is  made 
by  varying  the  distance  between  the  two  gold  electrodes  until 
the  galvanometer  shows  no  deflection.  At  the  final  setting  one  of 
the  potential  electrodes  should  be  oscillated  thru  a  small  amplitude 
in  the  direction  of  the  fall  of  potential  so  the  small  galvanometer 
deflections  are  equal  on  both  sides  of  the  zero.  This  setting 
made,  the  current  is  read  upon  the  ammeter,  also  the  distance 

*  Transactions  of  the  American  Institute  of  Electrical  Engineers,  February 
28,  1902,  page  827. 


248 


MEASURING  ELECTRICAL  RESISTANCE      [ART.  1123 


between  the  electrodes.     If  E  is  the  E.M.F.  of  the  standard  cell 
and  I  the  current  read  on  the  ammeter  then  the  resistance  be- 

TjJ 

tween  the  electrodes  is  R  =  -y-     If  I  is  the  distance  between  the 

electrodes  and  S  the  cross-section  of  the  tank  the  specific  resist- 
ance of  the  electrolyte  is 

SR      SE 


P=~T   =17 


(1) 


The  author  has  not  tried  this  method  and  so  cannot  speak  from 
personal  experience  as  to  its  accuracy  and  value.  Further  details 
may  be  obtained  by  reference  to  Dr.  Bering's  original  paper. 

1123.  The  Substitution  Method.  —  The  principle  of  this 
method  has  already  been  described,  par.  206,  Fig.  206b.  The 
apparatus  shown  in  Fig.  1120a  is  the  same  in  principle  as  that 
shown  in  Fig.  206b  and  may  be  used  for  the  measurement.  The 
substitution  method  of  measuring  electrolytic  resistances  is  inferior 
to  alternating-current  methods  but  may  be  used  with  advantage, 
perhaps,  for  quick  and  rough  determinations  of  the  resistivities 

of  electrolytes  in  battery  jars, 
electroplating  tanks,  etc.  The 
apparatus  required  is  generally 
at  hand  and  the  tube  or  cell 
may  be  submerged  in  any  body 
of  electrolyte  and  the  measure- 
ment be  very  quickly  made  in 
situ. 

If  the  electrolyte  is  a  silver, 
copper  or  nickel  solution  it  is 
well  to  use  electrodes  of  these 
metals.  For  other  solutions  gold 
or  platinum  electrodes  are  more 
suitable.  It  is  recommended  to 
make  the  electrodes  of  fine  wire- 
mesh,  as  bubbles  from  polariza- 
tion will  more  readily  escape 
from  the  electrodes. 

1124.  Resistance  of  "Grounds" 
(Bell  Telephone  Method).— 


-Moulding  Strip  for 

Ground  Wire. 
No.GB.W.G.  Iron  Wire. 


^Ground  Line 


Rocky  Soil 


l"  Commercial 
Pipe,  G'  long. 


FIG.  1124a. 


There  is  used  in  telephone  practice  what  is  termed  a  "  Cable  pro- 
tector ground."     These  are  ground   connections  made   at   tele- 


ART.  1124] 


RESISTANCE  MEASUREMENTS 


249 


phone  poles  to  afford  protection  for  aerial,  as  well  as  underground 
cable  plants,  against  lightning.  The  ground  connection  is  carried 
to  an  "  open  space  cutout,"  and  is  made  in  the  manner  shown  in 
Fig.  1124a.  It  is  necessary  that  the  resistance  of  such  grounds 
should  not  exceed  a  proper  limit  and  tests  and  reports  are  fre- 


K- 


FIG.  1124b. 

quently  made  upon  such  grounds.  The  method  adopted  by  the 
telephone  company  for  measuring  the  ground  resistance  is  known 
as  the  "-three  ground  method"  the  principle  of  which  may  be 
explained  as  follows: 

Referring  to  Fig.  1124b,  GI,  G2,  G3  are  three  ground  connections, 
having  resistances  to  ground  x,  y,  z,  respectively.     The  ground  GI 


No.  128    Receivers  on 
Double  Head.Band 


FIG.  1124c. 


is  the  permanent  ground.  It  is  required  to  determine  the  resist- 
ance x  of  this.  The  grounds  G2  and  G3  are  auxiliary  or  temporary 
grounds  which  are  constructed  in  order  to  effect  the  measurement. 
First  measure  (by  the  principle  of  the  method  shown  in  Fig. 


250  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1124 

1120c)  the  resistance  between  GI  and  G2,  then  between  GI  and  G3, 
and  lastly  between  G2  and  G3.  Calling  a,  6,  and  c  these  resistances 
respectively,  we  have 

x  +  y  =  a, 

x  +  z  =  b, 
and  y  +  z  =  c. 

From  these  three  equations  we  derive 

*  =  ^p.  CD 

The  exact  manner,  as  applied  by  the  telephone  company,  of 
making  the  measurements  is  clearly  explained  by  the  diagram 
shown  in  Fig.  1124c.  The  ground  resistance  may  vary  between 
such  limits  as  1  and  1000  ohms. 


CHAPTER  XII. 
ELEMENTARY  PRINCIPLES  OF  FAULT  LOCATION. 

1200.  Fault  Location.  —  All  electric  lines,  whether  used  for 
the  transmission  of  intelligence  or  power,  are  subject  to  what  is 
technically  designated  "  faults."  These  faults  consist,  in  general, 
of  a  complete  breaking  down  or  a  serious  deterioration  of  the 
insulation  of  the  line,  or  of  a  break  in  the  conductor.  If  the  defect 
develops  at  a  definite  point  it  becomes  important  to  be  able  to 
locate  its  place,  as  determined  in  distance  from  some  station  along 
the  line.  If  the  location  of  a  fault  can  be  quickly  and  accurately 
effected,  the  time  and  expense  of  making  a  repair  is  greatly  reduced. 

The  methods  which  have  been  developed  for  locating  faults 
from  a  station  on  the  line  chiefly  embody  some  form  of  resistance 
measurement  and  are  carried  out  with  resistance-measuring  appa- 
ratus. A  description  of  these  methods  properly  belongs,  therefore, 
to  a  work  of  this  character.  The  full  development  and  application 
of  the  methods  when  applied  to  submarine  cables  in  service  is 
complex  and  extensive,  and  should  be  confined  to  works  devoted 
especially  to  this  phase  of  the  subject.  The  fundamental  prin- 
ciples of  fault  location  upon  land  lines,  however,  are  easily  under- 
stood and  may  be  properly  described  here.  In  many  cases  their 
application  is  quite  simple.  In  other  cases,  however,  the  condi- 
tions under  which  the  relatively  simple  principles  have  to  be 
applied  are  complicated  by  networks  of  conductors,  multiplicity 
of  faults,  earth  currents,  variations  in  the  size  of  wires  in  the  same 
circuit,  and  other  causes  which  become  at  times  very  puzzling. 
The  majority  of  faults  may  be  located,  however,  by  one  familiar 
with  the  fundamental  principles  and  moderately  practiced  in  their 
application.  We  proceed  to  classify  and  tersely  describe  these 
fundamental  principles.  For  detailed  descriptions  of  the  special 
apparatus  which  instrument  makers  have  developed  for  fault  loca- 
tion the  reader  must  be  referred  to  the  trade  publications  which 
advertise  and  often  very  fully  describe  this  class  of  apparatus. 

251 


252  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1201 

1201.   Faults  Occurring  on  Land  Lines. 

(a)  Grounds.  —  This  is  a  common  fault  which  is  a  partial  or 
complete  breaking  down  of  the  insulation  whereby  the  conductor 
becomes  connected  to  the  ground  or  to  the  sheath  of  the  cable. 
If  the  ground  connection  is  localized  at  a  single  point  the  fault 
may  be  definitely  located,  but  not  infrequently  a  ground  connec- 
tion occurs  at  two  or  more  points.     In  this  event  a  precise  location 
of  each  point  where  the  conductor  is  grounded  is,  in  general,  not 
possible. 

(b)  Crosses.  —  These  are  faults  in  which  two  or  more  conductors 
in  the  same  sheath,  or  on  the  same  pole  lines,  become  connected 
together  or  crossed.     As  in  the  case  of  grounds  this  fault  may 
occur  at  one  place  or  several  places.     In  the  former  case  the 
localizing  of  the  fault  is  easy  and  locations  are  usually  made  by  the 
same  methods  as  are  used  for  locating  grounds. 


It- 

A  i 

2l 


>€ 


FIG.  1201. 

(c)  Opens.  —  An  open  is  produced  when  the  conductor-circuit 
becomes  broken  or  open.     If  the  circuit  is  not  at  the  same  time 
grounded  or  crossed,  the  point  at  which  the  circuit  is  open  may 
usually  be  located.     The  methods  differ,  however,  from  those  used 
for  grounds  or  crosses. 

(d)  Inductive  crosses.  —  These  faults  consist  of  a  transposition 
in  telephone  cables  of  single  sides  of  adjacent  pairs  of  conductors. 
They  result  from  incorrect  cable-splicing.     This  fault  is  illustrated 
by  Fig.  1201. 

Here  one  wire  of  the  pair  A  is  transposed  with  one  wire  of  the 
pair  B  at  the  splice  S.  As  there  is  an  electrostatic  capacity  be- 
tween pairs  of  wires  as  represented  by  the  condensers  C\  and  (72 
shown  in  dotted  line,  conversation  might  be  carried  on  between 
Ta  and  ta,  likewise  between  Tb  and  4,  but  there  will  also  be  bad 
cross  talk  between  Tb  and  ta,  and  Ta  and  tb.  Workmen  often 
attempt  to  correct  this  fault  by  connecting  the  telephones  as  indi- 


ART.  1204]  PRINCIPLES  OF  FAULT  LOCATION  253 

cated  in  dotted  line,  but  the  cross  talk  remains.     A  special  method 
will  be  given  for  locating  the  position  of  an  inductive  cross. 

1202.  Problems  in  Fault  Location.  —  The  chief  problems  and 
tests,  treated  under  the  subject  of  fault  location,  may  be  sum- 
marized as  follows: 

(a)  Identification  of  faulty  wires. 

(b)  Determination  of  the  resistance  of  conductor  loops. 

(c)  The  location,  in  distance  from  a  station,  of  grounds,  crosses, 
opens  or  inductive  crosses,  on  telephone  or  telegraph  lines. 

(d)  Fault  locations  when  loops  are  made  up  of  wires  of  different 
sizes  and  lengths. 

(e) Insulation  resistance  tests  of  installed  and  uninstalled  cables. 
Here  either  the  insulation  may  be  defective  in  particular  places 
or  the  defective  insulation  may  be  distributed  along  the  conductor. 

(f)  The  location  of  grounds  or  crosses  in  heavy  power  cables, 
requiring  special  apparatus. 

(g)  The  location  of  grounds  or  crosses  in  very  heavy,  short, 
underground  cables.     A  special  method  is  here  required. 

(h)  Location  of  grounds  or  crosses  in  high-tension  cables  which 
are  subject  to  inductive  disturbance  from  parallel  alternating- 
current  lines. 

(i)  Location  of  faults  in  submarine  cables,  during  manufacture, 
test,  and  after  being  installed.  These  problems  are  special  and 
are  not  considered  here. 

1203.  Relation  of  Principles  to  Practice  in  Fault  Location.  — 
Fault  location  depends  upon  certain  fundamental  principles  which 
must  be  clearly  understood  for  intelligent  work.     They  should 
be  studied  before  any  consideration  is  given  to  the  details  of  specific 
apparatus  and  methods.     There  are,   however,  many  practical 
points  which  must  be  considered  in  the  successful  application  of 
the  fundamental  principles.     These  practical  points  and  famili- 
arity with  fault-locating  apparatus  are  best  acquired,  however, 
by  practice  and  experience  in  the  field.     The  principles  of  the 
subject,  therefore,  should  chiefly  concern  us  here  and  we  proceed 
to  their  elucidation. 

1204.  Location  of  a  Ground  upon  a  Single  Line  with  Only  an 
Earth  Return.  —  Single  lines  with  only  earth  return  are  found 
in  overhead  telephone  and  telegraph  lines  in  unsettled  country 
and  in  single  lines  laid  under  water.     The  two  methods  to  be 
given  are  considered  as  having  more  of  a  theoretical  interest  than 


254  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1204 

practical  value,  altho  they  work  out  very  well  with  artificial  lines 
made  up  in  a  laboratory. 

(a)  By  testing  from  each  end  of  line. 

Signal  from  A  to  B,  Fig.  1204a,  to  open  KI  and  k\,  assuming 
that  ground  is  not  so  bad  but  that  it  is  possible  to  drive  a  signal 


FIG.  1204a. 

through.  Measure  with  resistance-measuring  device,  k  being 
closed  and  K  open,  the  resistance 

x  +  y  =  R. 

Leave  K  open  and  open  k.  From  end  B,  with  ki  closed  and  KI 
open,  measure  resistance 

z  +  y  =  Ri  =  Rt  —  x  +  y 

where  Rt  is  the  total  resistance  of  the  line. 
Then  the  resistance  to  the  fault  from  end  A  is 

_  R  —  RI  +  Rt  /..  x 

~~2~ 

The  distance  to  the  fault  may  then  be  calculated  when  the  feet  per 
ohm  of  the  wire  is  known. 

The  difficulties  which  would  be  encountered  in  this  test  would 
arise  (a)  from  earth-currents;  (b)  from  variations  in  the  resistance 
y  while  the  test  is  in  progress,  and  (c)  from  the  resistance  of  y  at 
times  being  so  high  that  the  resistances  x  or  z  would  be  small  in 
comparison. 

The  method  recommended  for  measuring  the  resistances  R  and 
#1  is  the  " voltmeter  and  ammeter  method"  given  in  par.  209. 
If  the  measuring  current  sent  over  the  line  is  large  it  will  give 
the  double  advantage  of  making  the  effect  of  earth  currents 
relatively  small  and  of  breaking  down  the  ground  resistance  y 
to  a  low  value.  The  method  should  prove  most  useful  and  rela- 
tively easy  of  application  in  a  wild  country  free  from  disturbing 
earth  currents  and  when  the  single  line  is  of  great  length. 


ART.  1204] 


PRINCIPLES   OF  FAULT  LOCATION 


255 


The  method  may  be  used  to  locate  a  grounded  point  in  a  coil 
of  insulated  conductor  in  a  tank,  or  when  wound  upon  a  metal 
frame.     In  this  case,  however,  both  ends  of  the  conductor  would 
be  at  the  same  place  and  the  location  could  be  made  more  easily 
and  accurately  by  the  Murray-loop  test  given  in  par.  1205. 
(b)  By  testing  from  one  end  of  line  only.     (Author's  method.) 
It  is  of  theoretical  interest  to  show  that  a  ground  may  be  located 
from  one  end  only  of  a  long,  single  conductor  line  when  the  only 
resistance  known  is  that  of  the  line  itself. 


^^^^ 


II 


u  ---  v  ---  H 


---  v  ----  > 


IL 


FIG.  1204b. 

The  apparatus  required  is  two  voltmeters,  a  known  resistance 
(preferably  one  which  can  be  given  different  values),  and  a  source 
of  steady  E.M.F. 

The  fundamental  assumptions  made  are:  (a)  that  the  total 
resistance  R  of  the  line  is  known;  (b)  that  the  ground  is  not  so 
bad  but  that  a  signal  may  be  driven  through  the  line  to  give 
directions  for  earthing  and  unearthing  it  at  its  far  end  at  proper 
times,  and  (c)  that  the  ground  connection  is  located  at  one  point 
only  and  that  its  resistance  remains  steady  during  the  test. 


256  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1204 

Represent  the  line  and  connections  for  the  test,  Fig.  1204b,  at 
I  with  line  grounded  at  both  ends,  at  77  with  far  end  of  line  open, 
and  let  Ieq  and  II eq  represent  equivalent  circuits. 

First,  a  signal  is  sent  to  ground  the  line,  as  represented  at  7,  and 
the  voltmeter  readings  v\  and  v  are  made  with  the  two  voltmeters. 

Second,  a  signal  is  sent  to*  disconnect  the  line  at  the  far  end,  as 
represented  at  77,  and  readings  v\  and  v'  are  made  with  the  two 
voltmeters.  We  now  have  R  =  x  +  z,  the  total  resistance  of  the 
line.  The  auxiliary  resistance  r  is  known  and  Vi,  Vi,  v}  and  vr  are 
measured  quantities.  We  then  have  (case  7  or  Ieq) 

z  =  R-x,  (1) 

^i  /     ,      zy  \  /o\ 

<vt   I        ~T~  '       T~~   I  '  \    / 

and  (case  77  or  IIeq) 

v'  =  V-f(x  +  y).  (3) 

In  these  three  equations  there  are  only  the  three  unknown  quan- 
tities x,  y,  and  2;  hence  it  is  possible  to  find  the  value  of  each  of 
them.  The  value  of  x,  the  resistance  of  the  line  from  q  to  p,  may 
be  obtained  as  follows: 

For  brevity,  write    •-  =  C     and      —  =  C' . 

These  are  the  currents  in  the  line  in  cases  7  and  77  respectively. 
Then,  from  Eq.  (2), 

=    z(Cx-v) 

and  from  Eq.  (3) 


Replacing  z  in  Eq.  (4)  by  its  value  R  —  x  we  obtain 

(R  -  x)  (Cx  -  v)      =  v'-  C'x 
v-Cx-C(R-x)  C' 

This  is  a  quadratic  equation  in  x.     Its  solution  is 

=  v_     ./v2      v'CR-C'Rv-vv' 
*^       Cf      \  C12  C*fC* 

(See  Appendix  II,  7,  Eq.  16.) 


(6) 


Since  from  Eq.  (2)  x  —  -^  — 


zy 


,  we  see  that  in  Eq.  (7)  the 


negative  sign  should  be  used  before  the  radical. 


ART.  1204] 


PRINCIPLES   OF  FAULT  LOCATION 


257 


If  in  Eq.  (7)  we  replace  C  and  C'  by  their  values  —  and  - 
respectively,  and  use  the  negative  sign  before  the  radical,  we  obtain 


Tf 

•fv    /      /  /\  / 

—  (V  Vi  —  VVi  )  —  V  V 


(8) 


Eq.  (8)  gives  the  resistance  of  the  line  to  the  point  where  it  is 
grounded  in  terms  of  voltmeter  readings  and  known  resistances. 
Two  important  simplifications  may  be  made: 

(a)  The  resistance  r  can  be  chosen  equal  to  the  total  resistance 
R  of  the  line. 

(b)  The  current  flowing  can  be  maintained  the  same  (by  means 
of  a  small  rheostat  in  circuit  with  the  battery  on  its  grounded  side) 
when  the  connections  are  changed  from  /  to  //.     In  this  case 


Assuming  that  conditions  (a)  and  (b)  are  fulfilled,  Eq.  (8)  reduces 


to 


(9) 


It  should  be  possible  to  always  meet,  in  practice,  these  two  re- 
quirements for  simplicity,  and  occasions  might  arise  where  the 
method  would  yield  practical  results  of  importance. 

The  method  was  tested  in  the  laboratory  as  follows:  470  feet 
of  No.  17  soft  iron  wire  were  strung  around  the  sides  of  a  room. 
A  fault  was  made.  Its  position  was  varied  and  also  the  resistance 
y  of  the  fault  to  earth,  and  locations  made  with  each  arrange- 
ment. 

The  total  resistance  of  the  wire  was  13.8  ohms  and  the  resist- 
ance r  was  given  this  value.  Locations  were  generally  made  with 
an  error  of  less  than  5  feet  in  335  feet. 

A  sample  set  of  readings  and  the  calculation  of  the  value  of  x 
are  given  below: 


Voltage  drop 
over  r, 
»i 

Voltage  drop  over 
x  +  z  and  y  in 
parallel, 

I 

Voltage  drop 
over  x  +  y, 
v' 

Calculated  value 
of  x  in  ohms  and 
feet 

True  value  of  x  in 
ohms  and  feet 

2.000 

1.886 

3.740 

9  .  84  ohms 
335  1    feet 

9  .  87  ohms 
336  2    feet 

258 


MEASURING  ELECTRICAL  RESISTANCE      [ART.  1205 


The  calculation  made  by  Eq.  (9)  is 
x  =i|^|"l.886  -  V(3.740  -  1.886)  (2.000  -  1.886)1  =  9.84  ohms. 

1205.  Loop  Methods  for  Locating  Grounds  or  Crosses.  —  In 
most  cases  where  a  fault  is  to  be  located,  other  wires,  not  faulty, 
run  between  the  same  points  as  the  faulty  wire,  and  one  or  more 
of  these  good  wires  may  be  used  in  making  a  test.  Good  wires 
are  usually  to  be  found  in  the  same  lead  sheath  with  the  faulty 


FIG.  1205a. 

wire.  An  end  of  a  good  wire  may  be  connected  with  an  end  of 
the  faulty  wire.  When  a  good  wire  and  a  faulty  wire  are  so  con- 
nected, the  combination  forms  a  loop.  Hence  the  name,  where 
this  combination  is  used,  of  "loop  test." 

The  "  Murray  loop  "  and  the  "  Varley  loop  "  are  the  two  loop 
tests  best  known  and  most  frequently  employed. 

The  Murray  Loop.  —  In  this  method  two  parts  of  the  conductor- 
loop  make  up  two  arms  of  a  Wheatstone  bridge.  The  other  two 
arms  are  obtained  from  the  testing  apparatus.  The  testing  appa- 
ratus usually  employed  is  a  portable  Wheatstone  bridge  with  keys, 


ART.  1205J 


PRINCIPLES  OF  FAULT  LOCATION 


259 


a  galvanometer,  and  several  cells  of  small  dry  battery  mounted  in 
the  same  case  with  the  coils.  The  rheostat  arm  is  made  variable, 
in  some  constructions  with  plugs  and  in  others  with  dials.  Appa- 
ratus of  this  character  is  popularly  known  as  "  A  portable  testing 
set."  The  oldest,  simplest,  and  least  expensive  type  of  portable 
testing  sets,  thousands  of  which  are  in  use,  is  shown  in  Fig.  1205a. 
The  more  expensive,  but  far  more  convenient,  sets  are  made  with 
decade  rheostats,  some  operated  with  plugs  and  others,  better 
still,  with  dials. 


FIG.  1205b. 

The  arrangement  employed  in  the  Murray-loop  test,  when  the 
testing  apparatus  is  a  Wheatstone  bridge  with  a  variable  rheostat, 
is  shown  as  a  theoretical  diagram  in  Fig.  1205b. 

Here/J  is  the  faulty  wire;  in  this  case  the  fault  is  indicated  as 
a  ground  at  the  point  F. 

eJ  is  the  return  wire,  which  is  free  from  a  fault.  The  "  loop  " 
is  eJf.  The  two  arms  of  the  bridge,  made  by  the  conductors  of 
the  loop,  are  eJF  and  Ff,  or  x. 

The  two  arms,  made  by  the  testing  apparatus,  are  B  and  R. 
In  this  application  of  the  method  the  arm  B  is  given  a  fixed  value, 
and  the  arm  R  is  varied  to  obtain  a  balance. 

The  galvanometer  Ga  is  joined  between  the  points  e  and  /,  and 
the  battery  has  one  terminal,  either  positive  or  negative,  joined 
between  the  arms  B  and  R  and  the  other  terminal  to  the  earth. 
The  relative  positions  of  the  galvanometer  and  battery  should 
always  be  chosen  as  above;  because,  if  the  galvanometer  is  not 
joined  to  the  earth,  it  will  not  be  affected  by  earth  currents. 
There  would  be  a  very  disturbing  factor  if  the  galvanometer  were 
put  to  earth. 

(a)  Call  r  the  resistance  of  the  loop  eJf  and  assume  this  as 
previously  known  or  obtained  by  a  measurement  at  the  time  of 
the  test. 


260  MEASURING  ELECTRICAL  RESISTANCE       [ABT.  1205 

Then,  if  the  resistance  B  is  maintained  fixed  and  R  is  varied  until 
the  bridge  is  balanced,  we  have 

B  _  r-x 

R~      x 
whence, 


FIG.  1205c. 

(b)  If  the  resistance  of  the  wire  is  uniform  throughout  its 
length  the  resistance  will  be  proportional  to  the  length.  Then,  if 
L  equals  the  total  length  of  the  loop,  in  any  chosen  units  of  length, 
and  d  is  the  distance  to  the  fault,  expressed  in  the  same  units  of 
length,  we  have 

B      L-d 

R~      d     ' 
whence, 


When  the  loop  is  made  up  of  wires  of  known  different  lengths  of 
different  sizes,  these  can  be  reduced  by  calculation  to  equivalent 
lengths  of  one  of  the  wires  of  the  loop  (see  §  1210). 

In  using  formula  (1)  in  order  to  calculate  the  distance  to  the 
fault,  the  resistance  of  the  wire  per  1000  feet  must  be  obtained 
from  a  wire  table  (for  copper  wire,  see  Appendix  III,  1).  A  slight 
error  in  gauging  the  diameter  or  in  estimating  the  temperature 
of  the  wire  may  cause  a  considerable  error.  If  the  length  L  of 
the  loop  is  known,  so  that  formula  (2)  may  be  used,  the  distance 
to  the  fault  may  be  obtained  with  much  greater  accuracy,  as  all 
errors  in  wire-size,  temperature,  and  specific  resistance  eliminate. 

An  arrangement  employed  for  the  Murray-loop  test,  when  some 
type  of  slide-wire  bridge  is  used  for  the  testing  apparatus,  is  shown 
in  Fig.  1205c. 


ART.  1205] 


PRINCIPLES  OF  FAULT  LOCATION 


261 


Here  the  faulty  wire  fj  is  shown  crossed  with  another  wire 
gh  at  F.  In  this  case  one  terminal  of  the  battery  is  connected 
to  the  crossed  wire  instead  of  being  put  to  the  earth.  Otherwise 
there  is  no  difference  in  procedure  for  locating  a  cross  or  a  ground. 

The  slide  wire  of  the  bridge  is  ef  and  the  balance  is  obtained 
by  moving  the  contact  p  along  the  slide  wire,  thus  varying  the 

ratio  -  • 
a 

(a)  If  r  is  the  total  resistance  of  the  loop,  we  have 

b      r  —  x 


or 


x  = 


r. 


(3) 


FIG.  1205d. 

(b)  Or,  if  L  is  the  total  length  of  the  loop  and  d  is  the  distance 

to  the  fault, 

a 


d 


L. 


a  +  b 
It  is  usual  to  make  a  +  b  equal  to  1000  scale  divisions. 


case 


1000 


(4) 

\ 

In  this 
(5) 


Thus  suppose  L  =  5286  feet  and  the  scale  reading  is  a  =  236, 
then  the  distance  to  the  fault  is 

d  =  0.236  X  5286  =  1247.5  feet. 

(c)  It  is  customary  in  order  to  gain  the  advantage  of  a  slide 
wire  of  double  length  to  employ  the  following  modification  of  the 
above  method: 

In  Fig.  1205d,  w  is  an  extension  of  the  slide  wire.  This  exten- 
sion is  a  wire  equal  in  length  and  size  to  the  slide  wire  or  it  is 
a  spool  resistance  which  is  made  exactly  equal  to  the  resistance 


262  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1205 

of  the  slide  wire  itself.     Then  the  relation  which  holds  for  a  bal- 
ance of  the  bridge  is 

w  -\-w  —  a      r  —  x 


From  this  relation 


*  =  -  5-  (6) 

w  2  v  ' 


The  quantities  a  and  w  are  resistances,  but  if  a  is  proportional  to 
lengths  read  upon  the  scale  of  the  slide  wire  and  if  this  scale  has 
1000  scale  divisions,  the  distance  to  the  fault  is 

d  =  15)0  2*  (7) 


Ba   •=: 

/    '       r 

FIG.  1205e. 

The  rule  for  locating  a  fault  by  this  arrangement  of  the  Murray 
loop  then  becomes:  Connect  the  faulty  wire  to  the  free  end  of  the 
slide  wire  and  the  good  wire  to  the  end  of  the  extension  resistance. 
Obtain  a  balance  and  read  the  number  of  scale  divisions  from  the 
end  of  the  slide  wire  joined  to  the  faulty  wire. .  This  scale  reading 
divided  by  1000  and  multiplied  by  one  half  the  length  of  the  loop 
is  the  distance  to  the  fault. 

This  modification,  if  used  without  corrections,  also  assumes  that 
the  resistance  of  the  wires  in  the  loop  is  uniform  and  proportional 
to  length  of  wire. 

The  Varley  Loop.  —  The  Varley  loop  differs  from  the  Murray 
loop  in  having  a  portion  of  the  resistance  of  the  loop  in  the  testing 
apparatus. 

The  Varley-loop  test  is  generally  made  with  a  portable  testing 
set.  It  is  most  conveniently  applied  when  the  ratio  arms  in  the 
set  can  be  given  ratios,  1,  10,  100,  1000,  etc. 

The  connections  are  shown  schematically  in  Fig.  1205e. 

Here  a,  b  are  the  ratio  arms  of  a  Wheatstone  bridge  (or  portable 


ART.  1205]  PRINCIPLES   OF  FAULT  LOCATION  263 

testing  set)  which  are  set  at  a  fixed  ratio  which  we  shall  call 

^  =  D.  R  is  the  rheostat  of  the  set  which  can  be  varied,  as  at  p, 
o 

by  plugs  or  dials.     The  complete  loop  is  now  eJpf,  and  the  fault, 
a  ground  or  cross,  is  at  F. 
For  a  balance 

a       n       r~x 

b  =        -fTTx' 

where  r  is  the  resistance  of  the  conductor  loop  ejp.     From  this 

relation 

r  -  DR  ,  . 

(a)  x=-^+I' 

For  unity  ratio,  or  D  =  1, 

r-R 
x-    -r, 

or 

|  =  I  -  x  =  Ktd,  (9) 

where  d  is  now  the  distance  to  the  fault  reckoned  from  J,  the  point 
of  union  of  the  good  and  bad  wires,  and  KI  is  a  constant  of  pro- 
portionality. If  the  wire  in  the  loop  is  uniform  and  the  faulty 
and  good  wires  of  equal  cross-section,  then, 

I  =  tf  A  do) 

where  I  is  the  length  of  one  wire. 
From  Eqs.  (9)  and  (10) 

(b)  d  =  |j.  (11) 

Thus  the  rule  to  follow  in  making  a  test  becomes:  Set  the  ratio 
arms  of  the  bridge  at  unity  ratio.  Vary  the  rheostat  until  a 
balance  is  obtained.  Then  the  distance  to  the  fault,  reckoned 
from  the  far  end  of  the  loop,  is  the  ratio  of  the  resistance  of  the 
rheostat  to  the  resistance  of  the  conductor  loop  multiplied  by  the 
length  of  one  wire. 

1  (c)  The  gauge  of  the  wire  but  not  its  length  may  be  known. 
In  this  case  we  have  the  length  of  the  conductor  loop,  or  2  Z, 
proportional  to  the  resistance  of  the  conductor  loop,  or  2  I  =  Kr, 

or  i  =  K. 


264  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1205 

Here  K  is  a  constant  which  expresses  feet  per  ohm  or  meters 
per  ohm.  Knowing  the  gauge  and  kind  of  wire,  the  value  of  this 
constant  may  be  obtained  from  a  wire  table.  Putting  in  Eq.  (11) 
the  above  value  of  I,  the  distance  to  the  fault  reckoned  from  / 
becomes 

d  =  R~  (12) 

If  K  is  given  as  feet  per  ohm,  the  distance  in  feet  to  the  fault 
from  the  far  end  is  the  product  of  the  rheostat  resistance  in  ohms 
and  one  half  the  number  of  feet  per  ohm  of  the  particular  size  of 
wire  in  the  loop. 

The  relation  given  in  Eq.  (12)  is  one  which  is  very  commonly 
used  by  telephone  line  testers. 

(d)  Another  relation  sometimes  used  is  obtained  thus:    for  a 

balance, 

a       r  —  x 


br  —  aR 

*  •  T+F- 

Eq.  (13)  expresses  the  resistance  to  the  fault,  reckoned  from 
the  end  of  the  line  where  the  test  is  made. 

Another  method  of  using  the  Varley-loop  test  is  applied  as 
follows  : 

(e)  From  Eq.  (8)  we  found  when  D  =  1  that  x  =  -      —  .     Here 

Zi 

again,  if  the  gauge  and  kind  of  wire  is  known,  we  have  the  dis- 
tance to  the  fault,  reckoned  this  time  from  the  instrument  end, 

^        r  —  R  .„ 
s  =  Kx  =  —  x  —  K. 
A 

If  K  is  given  as  feet  per  ohm  the  distance  in  feet  to  the  fault  from 
the  instrument  end  is 

s  =  —^  —  X  feet  per  ohm.  (1.4) 

Illustrative  Examples. 

(1)  Illustration  of  (a),  Eq.  (8).     The  measured  resistance  r  of 

the  loop  was  32  ohms.     The  ratio  T  =  D  used  was  0.1,  and  a 
balance  was  obtained  with  a  rheostat  setting  of  184  ohms.     Then 


ART.  1206]  PRINCIPLES  OF  FAULT  LOCATION  265 

by  Eq.  (8)  the  resistance  to  the  fault  from  the  instrument  end  was 

x  =  32  -O-^X  184  =  12.36  +  ohms. 

(2)  Illustration  of  (b),  Eq.  (11).     The  total  resistance  r  of  a 
cable  loop  was  267  ohms.     The  length  of  one  wire,  or  the  cable, 
was  40,650  feet.     The  resistance  in  the  rheostat  which  gave  a 
balance  was  R  =  27  ohms.     Hence  by  Eq.  (11)  the  distance  to 

the  fault  was 

97 
d  =  ~  X  40,650  =  4111  feet. 

(3)  Illustration  of  (c),  Eq.  (12).     In  a  circuit  the  part  of  the 
faulty  wire  from  where  the  wires  were  joined  was  composed  of 
No.  14  copper  wire.     A  balance  was  obtained  with  25  ohms  in  the 
rheostat.     By  a  wire  table  it  is  found  that  No.  14  wire  has  396.6 
feet  per  ohm.     Then  the  distance  from  /  to  the  fault  is 


d  =  25        !    =  4957.5  feet. 

2i 

(4)  Illustration  of  (e),  Eq.  (14).  The  resistance  of  a  loop  meas- 
ured 74  ohms,  and  a  balance  was  obtained  with  the  rheostat  set  at 
23  ohms.  The  size  of  the  wire  in  the  loop  was  No.  14,  which  has 
396.6  feet  per  ohm.  Then  by  Eq.  (14)  the  distance  to  the  fault 
from  the  instrument  end  was 

74  _  oq 
8  =  •^-^-  X  396.6  =  10113.3  feet. 

1206.  Notes  on  the  Varley  Test.*  —  "  This  test  is  extremely 
useful,  particularly  on  multiplied  telephone  cables.  By  multi- 
plied one  is  to  understand  that  the  same  pair  of  wires  is  tapped 
into  a  number  of  different  terminals,  as  shown  in  Fig.  1206. 
This  pair  is  multiplied  at,  four  points,  A,  B,  C,  and  D. 

When  the  arms  of  the  Wheatstone  bridge  are  made  even  in  the 
Varley  test,  the  formula  R  =  r  —  2  x  means  that  the  resistance 
in  the  rheostat  when  balance  is  obtained  is  equivalent  to  that  of 
both  sides  of  the  pair  from  the  end  where  the  helper  makes  his 
connection  to  the  fault. 

It  is  an  easy  matter  for  the  tester  to  memorize  the  constants 

*  These  notes  were  written  by  J.  W.  Wright,  of  the  Bell  Telephone  Com- 
pany of  Pennsylvania  for  The  Leeds  and  Northrup  Company,  and  with  this 
company's  permission  are  here  reproduced. 


266 


MEASURING  ELECTRICAL  RESISTANCE       [ART.  1206 


for  the  number  of  feet  per  ohm  of  the  three  or  four  most  common 
sizes  of  wire  used  for  telephone  cables. 

Knowing  the  gauge  of  the  wire  it  is  merely  necessary  to  multiply 
one  half  the  number  of  feet  per  ohm  of  that  size  wire  by  the 
rheostat  reading  in  order  to  get  the  distance  to  the  fault  in  feet. 


FIG.  1206. 

The  application  of  this  to  multiplied  cable  may  be  readily  shown. 
Assume  a  fault  at  F.  The  tester  is  at  Ex  and  the  helper  connects 
the  two  wires  together,  say  at  C.  The  Varley  test  then  gives 
the  ohms  back  from  C  to  the  apparent  location  of  the  fault  (in 
this  case  at  N  where  the  pair  to  B  is  tapped  on  the  main  cable). 
Having  the  resistance  from  C  to  Nt  a  rough  calculation  involving 
feet  per  ohm  multiplied  by  Varley  reading  gives  the  distance  from 
C  to  the  apparent  fault. 

From  the  cable  diagram  one  can  read  this  distance,  which  should 
be  about  that  between  C  and  N.  The  connection  at  C  is  then 
removed  and  replaced  on  the  same  pair  at  B  and  the  above  process 
repeated.  In  order  not  to  be  misled  by  the  variation  in  resistance 
due  to  changes  in  temperature,  it  is  well  for  the  tester  to  measure 
some  known  length  once  each  week  or  two  and  divide  distances  by 
resistance  to  obtain  the  proper  constant. 

That  this  is  quite  important  may  be  understood  from  the  fact 
that  for  underground  cable  the  constants  will  vary  10  per  cent 
between  summer  and  winter  temperatures. 

In  cases  where  there  are  cables  of  two  different  gauges  spliced 
together  it  is  easy  to  figure  out  the  location  without  making  any 
gauge  correction. 

For  instance,  consider  the  case  where  a  pair  of  wires  of  one  gauge 
is  attached  to  another  pair  of  equal  length  but  different  gauge,  the 
total  loop  resistance  being,  say,  40  ohms  and  the  Varley  test  show- 
ing a  balance  at  10  ohms  with  equal  bridge  arms. 


ART.  1207]  PRINCIPLES  OF  FAULT  LOCATION  267 

This  would  mean  that  the  fault  was  10  ohms  from  the  far  end. 
Knowing  the  two  gauges  one  can  estimate  mentally  if  this  amount 
of  resistance  will  carry  the  location  beyond  the  junction  point  of 
the  two  sizes.  If  not,  then  multiply  by  the  constant  for  the  gauge 
wire  on  the  far  end,  which  will  give  directly  the  distance  from  that 
end  to  the  fault. 

In  any  case  where  the  balancing  resistance  carries  the  location 
into  the  section  nearest  the  locator,  then  instead  of  multiplying  the 
constant  by  the  rheostat  reading,  subtract  this  reading  from  the 
total  loop  resistance  and  multiply  the  difference  by  the  constant 
for  the  gauge  wire  nearer  to  the  tester.  This  method  is  lengthy  of 
explanation  but  when  once  one  gets  the  idea,  these  processes  are 
mainly  mental  and  really  take  very  little  time. 

Tests  may  be  made  this  way  on  all  but  the  very  shortest  cables 
and  not  then,  for  the  reason  that  ordinary  bridge  sets  are  not 
subdivided  below  one  ohm  in  the  rheostat. 

Very  often,  however,  one  is  able  to  interpolate  proportionally  to 
the  deflection  of  the  galvanometer  when  a  close  approximation  is 
necessary. 

However,  when  it  comes  to  a  question  of  inches  on  a  short  length 
of  wire,  either  the  Murray  test  or  the  Varley  test  with  unequal 
bridge  arms  should  be  used  for  accuracy. 

The  formula  R  =  r  —  2  x  is  the  universal  Varley  formula  for 
equal  bridge  arms.  Knowing  the  gauge  of  the  bad  wire  it  is 
possible  to  obtain  the  location  by  solving  for  x.  When  a  good 
wire  of  large  size  is  used  and  the  fault  is  near  the  far  end  of  the 
smaller  wire  it  sometimes  happens  that  a  balance  cannot  be  ob- 
tained with  the  bad  wire  in  series  with  the  rheostat.  In  these 
cases  it  is  necessary  to  reverse  the  wires  on  their  respective  binding 
posts.  Then  balance  as  usual  and  use  the  formula  with  R  negative 
or  -  R  =  r  -  2  x, 

R  +  r  „ 

*-— y— 

1207.   Modified  Loop  Methods  to  Meet  Special  Conditions.  — 

Faulty  wire  of  known  length  and  two  good  wires  of  unknown  length 
and  resistance.     (H.  W.  Fisher's  method.) 

This  method  is  widely  applicable.  The  faulty  wire  may  be  an 
aerial  of  known  length  or  resistance,  or  a  wire  in  a  cable  of  known 
length.  The  two  good  wires  (either  aerials  or  wires  in  a  cable) 


268 


MEASUBING  ELECTRICAL  RESISTANCE       [ART.  1207 


must  meet  the  one  necessary  condition  of  terminating  at  the  same 
points  as  the  faulty  wire. 

The  meaning  of  the  symbols  is  apparent  from  an  inspection  of 
the  diagrams  I  and  II,  Fig.  1207a. 


(a)  For  connections  made  as  in  I  a  balance  of  the  bridge  is 
obtained  when 


a  x 

and  for  connections  made  as  in  II  a  balance  is  obtained  when 


w 


(2) 


Eliminating  y  from  Eqs.  (1)  and  (2)  the  resistance  from  point  /  to 
the  fault  F  is 

.  (3) 


ai  (a  +  6) 

Ordinarily  when  a  balance  is  obtained  in  case  II  the  arm  6  of  the 
bridge  would  not  be  changed,  in  which  case 

a  (a,  +  6) 


x  = 


-  7—     . 

(a  +  6) 


(4) 


ART.  1207]  PRINCIPLES  OF  FAULT  LOCATION  269 

(b)  This  method  is  more  simply  applied  in  the  slide-wire  type 
of  bridge.     In  this  case  the  point  of  attachment  p  of  the  battery 
is  moved  over  a  uniform  resistance  to  secure  a  balance.     Then 
the  resistance  a  +  6  =  ai  +  61  at  all  times,  and  Eq.  (3)  becomes 

a  /rx 

x  =  —  w.  (5) 

ai 

If  the  length  I  of  the  faulty  wire  is  known  and  d  is  the  distance 
from/  to  the  fault  F  Eqs.  (3),  (4),  and  (5)  may  be  written 

,      a  (ai  +  61)  j  /->. 

-*(a  +  V)1 
a  (QI  +  6)  7  /7x 

d  =  ^+vl  (7) 

d-±l.'  (8) 

Ol 

(c)  This  method  may  be  modified  as  follows: 

1st.  Join  the  two  good  wires  at  their  distant  end  and  ground 
them  at  the  point  of  connection. 

2d.   Measure  the  resistance  of  the  loop  so  formed.     Call  z-\-y  =  r. 

3d.  By  means  of  a  Murray  or  Varley  test  ascertain  the  indi- 
vidual resistance  of  each  wire.  Thus  z  +  y  =  r  and  =  T  >  whence 

b  a 

z  = —  :—=•  r  and  y  =  -  r~rr- 
a  +  b  a  +  b 

4th.  Connect  at  the  far  end  one  of  the  good  wires  with  the 
faulty  wire  and  measure  the  total  resistance  of  the  loop.  Then 
obtain  the  resistance  of  the  faulty  wire  by  subtracting  the  resist- 
ance of  the  good  wire  from  the  resistance  of  the  loop. 

5th.  The  resistance  of  the  faulty  wire  now  being  known,  the 
distance  to  the  fault  is  determined  by  either  the  Murray  or  Varley- 
loop  method, 

(d)  The  Fisher  method  described  above  is  very  conveniently 
applied  when  the  length  of  the  cable  which  contains  the  faulty  wire 
is  known,  but  not  necessarily  the  length  of  the  faulty  wire  itself. 
The  faulty  wire  may  twist  in  the  cable  and  therefore  be  longer 
than  the  cable  sheath.     If  this  twist  is  uniform  thruout  the  length 
of  the  cable,  the  method  correctly  locates  the  fault  as  a  certain 
distance  measured  along  the  cable  sheath  from  the  tester's  end. 


270 


MEASURING  ELECTRICAL  RESISTANCE       [ART.  1207 


Furthermore,  the  slide  wire  of  the  bridge  may  have  an  extension 

resistance  q  of  any  value  upon  one  end. 

This  modification  may  be  briefly  described  as  follows: 

In  I  and  II,  Fig.  1207b,  q  is  the  extension  resistance,     b,  a,  bi,  0,1, 

y,  w,  and  x  are  resistances.     K  and  k  are  constants.     Z  is  the  length 


!- ~kx=d  ~        Hf7 

II 
FIG.  1207b. 

of  the  cable  sheath  and  d  is  the  distance  to  the  fault  F.     With 
the  connections  as  in  I, 

q  +  b  =  y  +w  —  xt 

a  x 

With  the  connections  as  in  II, 


w 


(9) 
(10) 


Whence,  from  Eqs.  (9)  and  (10), 


But 
hence, 

Also, 
hence, 


q  +  ai  +  61  =  q  +  a  +  6; 


a 
x  =  —  w. 


d 
T 

ft/ 

° 
Oi 


(ID 


ART.  1208]  PRINCIPLES   OF  FAULT   LOCATION  271 

Thus  the  distance  to  the  fault  equals  the  ratio  of  the  two  readings 
multiplied  by  the  length  of  the  cable  sheath.  By  using  an  exten- 
sion resistance  of  the  right  magnitude,  a  point  can  always  be  found 
upon  the  bridge  wire  in  case  II  which  will  give  a  balance  whatever 
may  be  the  resistance  of  the  wire  y.  When  y  =  w  and  q  =  the 
resistance  of  the  bridge  wire,  the  greatest  length  possible  of  the 
bridge  wire  will  be  utilized.  By  using  an  extension  resistance  q 
greater  precision  may  be  secured. 

The  general  method  of  par.  1207  is  very  usefully  applied  for 
locating  faults  in  large  cables  which  are  wound  upon  reels  and  lie 
in  the  factory  waiting  to  be  tested.  One  end  only  of  the  cable  is 
brought  into  the  testing  room.  Two  wires,  no  special  regard  being 
given  to  their  resistance,  lead  from  the  testing  room  and  are  con- 
nected to  the  other  end  of  the  cable.  The  distance  to  the  fault  is 
then  obtained  from  the  simple  relation  given  in  Eqs.  (8)  and  (11). 

1208.  Where  the  Faulty  Wire  is  of  Known  Length  and  there 
is  Only  One  Good  Wire  of  Unknown  Length  and  Resistance.  — 
There  are  two  methods  of  finding  the  resistance  or  the  distance  to 

Q  /Return  Wire  of  Unknown  Resistance  y 

b. 


K kx=d HF 


H -kw=l -+- 1 


Return  Wire  of  Unknown  Resistance  y 


kx=d 


"1 

* kw=l  — 

II 
FIG.  120Sa. 

the  fault  in  this  case.  The  first  method  described  is  well  known. 
It  requires  that  a  test  be  made  from  each  end  of  the  faulty  wire. 
The  second  method  described  is  the  author's  method  and  by  its 
means  the  fault  may  be  located  by  testing  from  one  end  only. 

(a)  Testing  from  both  ends  of  faulty  wire. 

1st.  Join  the  faulty  wire  and  good  wire  together  at  end  B,  as  in 
I,  Fig.  1208a,  and  make  connections  with  the  testing  apparatus  as 
shown  in  the  diagram.  For  precise  work  it  is  necessary  to  consider 


272  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1208 

the  influence  on  the  result  of  the  lead  wire  c  which  joins  the 
point  /  to  an  end  of  the  faulty  wire.  Obtain  a  balance  by  varying 
the  bridge  arms.  Let  the  resistance  values  obtained  be  a  and  b. 

2d.  Join  the  faulty  and  good  wire  together  at  end  A,  as  in  II, 
Fig.  1208a,  and  complete  the  connections  to  the  testing  apparatus 
at  end  B,  using  the  same,  or  an  equivalent,  lead  wire  c.  Obtain 
a  second  balance  of  the  bridge.  Let  ai  and  61  be  the  resistance 
values  which  give  a  balance  this  time. 

Let  I  =  kw  =  the  length  of  the  faulty  cable,  where  w  =  the 
resistance  of  the  faulty  wire  in  the  cable  and  A;  is  a  constant. 

Let  d  =  kx  =  the  distance  to  the  fault  from  end  A,  where  x  = 
the  resistance  of  the  faulty  wire  from  end  A  to  the  fault. 

Let  y  =  the  unknown  resistance  of  the  good  wire. 

Let  c  =  the  resistance  of  the  lead  wire  used. 

The  resistance  of  the  wire  to  the  fault  is  now  obtained  as  follows : 
In  case  I, 

.    b _y+w-x 

a~       x+c 
In  case  II, 

h  — v  +  *.  (2) 

«i      w  —  x  +  c 
Eliminating  y  from  Eqs.  (1)  and  (2),  we  finally  obtain 

a  (at +61)  o&i-6ai 

a  (ai  +  61)  +  ai  (a  +  6)  W  ^  a  (ai  +  61)  +  ai  (a  +  b) 

With  the  slide-wire  type  of  bridge,  we  would  have  a  +  b  =  ai  +  61; 
in  which  case 

a  obi  —  ba\  ... 

x  =  —     —  w  +  7 —  r~ LW —    — T  c-  (4) 

a  +  ai          (a  +  6)(a  +  ai) 

Eqs.  (3)  and  (4)  show  that  for  exact  work  the  resistance  c  cannot 
be  neglected  unless  it  is  made  small.  We  shall  assume  that  this 
can  always  be  done  so  that  it  is  only  necessary  to  use  the  first  term 
of  the  right-hand  member  of  Eq.  (3)  and  of  Eq.  (4). 

If  c  is  made  negligible,  and  I  the  length  of  the  cable  which 
contains  the  faulty  wire  is  known,  then,  as  resistances  are  pro- 
portional to  lengths,  Eqs.  (3)  and  (4)  become 

7      a  (fli  +  61) 7  ,-<, 

•  aCai+fcO+aiCa+dr 

and  d  =  —^—  I  (6) 


ART.  1208] 


PRINCIPLES  OF  FAULT  LOCATION 


273 


(b)  Testing  from  one  end  of  faulty  wire.     (Author's  Method,  I.) 
In  this  method  a  known  auxiliary  resistance  P  is  required. 
When  a  slide-  wire  bridge  is  used  the  test  is  made  as  follows: 
With  the  connections  as  in  I,  Fig.  1208b, 


a         y  +  w 

where  s  is  the  length  of  the  bridge  wire  and  a  the  reading  from 
the  end  /,  both  in  scale  divisions. 


Return 


f-H^ 

N^ 

*_y 

K- 

L  ^ 
r-oxncGE 

rcKXKKXSKKXKrtxxKKKXXKxyxx  xyy 

1 

Tx~o~?n 

Return     Wire    ot_U^notvn 


s-a 


II 

FIG.  1208b. 

With  the  connections  made  as  in  II, 

s  —  ai  _  P  +  y  +w  —  x 
ai  x 

From  Eqs.  (7)  and  (8)  we  have,  by  eliminating  y, 


a  i 


s  —  a 


P. 


(8) 


(9) 


Eq.  (9)  gives  the  resistance  to  the  fault  F  from  the  end  where 
the  test  is  made. 

If  the  bridge  wire  has  a  length  of  1000  scale  divisions,  s  =  1000 
and 


X  = 


1000  -  a 


P. 


(10) 


274  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1208 

It  will  be  noted  that  in  this  method  it  is  not  even  necessary  to 
know  the  resistance  of  the  faulty  wire  to  obtain  the  resistance  to 
the  fault.  If,  however,  the  resistance  w  of  the  faulty  wire  is 
known  then  the  distance  to  the  fault  will  be 

I 

™' 

where  I  is  the  length  of  the  cable  which  contains  the  faulty  wire. 
The  auxiliary  resistance  P  should  be  given  the  same  order  of 
magnitude  as  the  resistance  of  the  wire  y  or  w. 
Example  taken  from  an  actual  test  to  illustrate  Eqs.  (10)  and 

(ID. 

The  length  of  the  faulty  cable  was  I  =  1252  meters. 
The  resistance  of  the  faulty  wire  was  w  =  8.34  ohms. 
P  =  13.2  ohms. 

a  =  399  scale  divisions. 
ai  =  111.7  scale  divisions. 

s  =  1000  scale  divisions. 
Therefore,  by  Eq.  (10), 


By  Eq.  (11)  the  distance  to  the  fault  was 


'2  x  3  -  2'453  x  3  -  368'2 


The  actual  distance  to  the  fault  was  afterward  determined  and 
found  to  be  374  meters.  This  makes  the  error  5.8  meters  or  a 
little  over  0.5  of  1  per  cent  of  the  length  of  the  cable. 

(c)  Testing  from  one  end  of  faulty  wire.     (Author's  Method,  II.) 

This  method  is  only  a  modification  of  I,  above.  It  will  be 
noted  that  the  method  may  be  applied  by  employing  the  same 
apparatus,  and  connections  similar  to  those  used  in  making  a 
Varley-loop  test. 

In  this  modification  the  ratio  arms  a  and  b  are  maintained  fixed 
and  a  balance  is  secured  by  varying  an  auxiliary  resistance  R. 
This  resistance  may  be  the  rheostat  of  the  portable  testing  set. 
Call  z  the  unknown  resistance  of  the  good  wire  plus  the  faulty 
wire  from  its  far  end  J  to  the  fault  F. 

Call  x  the  resistance  of  the  faulty  wire  from  the  near  end  /  to 


ART.  1208]  PRINCIPLES  OF  FAULT  LOCATION 


275 


the  fault  F.     Then  with  the  connections  as  in  I,  Fig.  1208c,  we 
have  for  a  balance,  obtained  by  varying  R, 

t-rb-  (12) 

With  the  connections  as  in  II,  we  have  for  a  balance,  obtained  with 
a  new  value  of  the  resistance  Ri, 


a  _  z  -{-  x 
b~  ~W 


For  brevity  call  the  ratio  T  =  N. 


(13) 


\  ^H^ 

F^V^~-. 

vvVWt                        x 

"W 

II 

FIG.  1208c. 

Then  from  Eqs.  (12)  and  (13)  we  obtain,  by  eliminating  z, 

N  (R!  -  R) 


If  the  ratio  T  =  N  is  made  unity,  then 


X  = 


I  —  R 


(14) 


(15) 


Eq.  (14)  or  (15)  gives  the  resistance  from  the  testing  end  to  the 
fault,  when  the  resistance  of  the  faulty  and  single  good  wire  are 
both  unknown.  This  method  illustrates  the  very  general  applica- 
bility of  the  Varley-loop  test  when  modified  to  meet  particular 
conditions.  Apparatus  suitable  for  a  Varley-loop  test  will  serve 
for  practically  all  cases  of  grounds  and  crosses  on  telephone  or 
telegraph  lines.  By  the  use  of  a  little  ingenuity  the  standard 


276 


MEASURING  ELECTRICAL  RESISTANCE      [ART.  1209 


Varley-loop  method  may  be  modified  to  take  care  of  such  special 
conditions  as  are  illustrated  in  the  method  above. 

1209.  One  Good  Wire  of  Unknown  Length  and  Two  Faulty 
Wires  Equal  in  Length  and  Resistance. 

(a)  Mr.  Henry  W.  Fisher's  Method.* 

It  sometimes  happens  that  all  of  the  wires  in  a  cable  become 
defective  and  it  may  be  difficult  to  secure  two  good  wires  and  apply 
the  test  given  in  par.  1207. 


b'           c 

b 

m 

n 

T 

FIG.  1209. 

If  one  good  wire  can  be  obtained  for  a  loop  test,  the  separate 
resistances  of  the  good  and  bad  wires  can  be  determined  by  a 
method  devised  and  used  by  Mr.  H.  W.  Fisher,  at  a  time  when  all 
the  conductors  in  a  cable  were  bad  and  there  was  only  one  aerial 
wire  available  for  the  test. 

Let  b  and  b'  represent  the  bad  wires  and  c  the  good  wire. 

Measure  the  combined  resistance  of  c  and  b  and  call  it  r. 

Measure  the  combined  resistance  of  c  and  b'  and  call  it  m. 

Measure  the  combined  resistance  of  c  in  series  with  6  and  bf  in 
multiple  and  call  it  n. 

Fig.  1209  shows  the  connections  and  underneath  each  is  the 
letter  designating  the  resistance. 

The  resistance  of  b  =  r  —  n  -f  \/n  (n  —  r  —  m)  -f  rm.  (1) 

The  resistance  of  b'  =  m  —  n  +  \/n  (n  —  r  —  m)  -\-  rm.  (2) 

Where  the  resistances  r  and  m  do  not  differ  by  more  than  2  per 

*  The  description  of  this  method,  as  first  used  by  Mr.  H.  W.  Fisher,  is 
taken  from  published  literature  of  The  Leeds  and  Northrup  Company,  with 
its  permission. 


ART.  1209]  PRINCIPLES   OF  FAULT  LOCATION  277 

cent  or  3  per  cent,  the  following  approximate  and  much  simpler 
equations  may  be  employed: 

b  =  -2n,  (3) 


This  method  has  given  good  results  where  used.  It  is  only 
strictly  applicable  when  the  bad  wires  b  and  b'  are  faulty  at  the 
same  point. 

Leading  wires  can  be  used  without  in  any  way  affecting  the 
result,  the  total  measured  resistance  of  each  loop  being  taken  as 
represented  by  the  letters  r,  m  and  n.  The  same  leading  wires, 
however,  must  be  used  throughout  the  tests. 

Having  thus  determined  the  resistances  of  the  faulty  wires,  a 
Loop  Test  can  be  applied  and  the  fault  located. 

(b)  When  there  are  one  or  more  faulty  wires  of  unknown  length  and 
resistance  and  only  one  good  wire  of  unknown  length  and  resistance. 

The  author's  methods  (b)  and  (c),  par.  1208,  can  be  applied 
generally  in  the  above  cases. 

Referring  to  I  and  II,  Fig.  1208b,  we  may  call  the  total  resist- 
ance w  of  the  faulty  wire  unknown.  Eq.  (9)  or  (10)  of  par.  1208 
gives  the  resistance  to  the  fault  in  terms  of  two  scale  readings  and 
the  auxiliary  resistance  P.  The  size  and  temperature  of  the  faulty 
wire  can  be  determined  and  then  the  distance  to  the  fault  can  be 
calculated  with  the  aid  of  a  wire  table.  The  wire,  however,  may 
twist  in  the  cable  and  be  longer  in  reality  than  the  cable  itself. 
The  tendency,  therefore,  would  be  to  place  the  fault  at  too  great  a 
distance  from  the  testing  end. 

The  unknown  resistance  w  of  the  faulty  wire  may  be  deter- 
mined if  a  duplicate  test  is  made  at  the  other  end  of  the  line.  In 
this  case  the  resistance  Xi  from  the  other  end  of  the  line  will  be 
given  in  the  same  manner  as  the  resistance  x  from  the  first  end  of 
the  line.  Then  x  +  x\  will  be  w,  the  total  resistance  of  the  line. 

IT 

If  I  is  the  length  of  the  cable,  —  ;  --  I  will  be  the  distance  measured 

'x  +  xi 

along  the  cable  to  the  fault  from  one  end,  and  —  ^  —  I  will  be  the 

x  +  xi 

distance  to  the  fault  measured  along  the  cable  from  the  other  end. 
A  test  of  this  character  would  become  useful  when  one  aerial  is 


278  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1210 

available  and  all  the  wires  of  unknown  length  in  a  cable  of  known 
length  have  become  grounded  at  some  one  point. 

12 10.   Methods   of  Applying  Corrections  in  Loop  Tests.  — 

(a)  When  the  good  and  faulty  wires  differ  from  each  other  in  both 
size  and  length,  but  where  the  size  and  length  of  each  is  known. 

If  the  two  wires  of  the  loop  differ  in  the  above  respects,  the  fault 
will  be  incorrectly  located  by  a  Murray  or  Varley-loop  test  unless 
a  correction  is  applied. 

To  make  this  correction,  multiply  the  length  of  the  good  wire 
by  its  rated  resistance  per  unit  of  length  and  divide  the  product 
by  the  rated  resistance  per  unit  of  length  of  the  faulty  wire,  add 
to  this  result  the  length  of  the  faulty  wire  and  call  the  final  result 
the  total  length  of  the  loop. 

In  applying  the  Murray  and  Varley-loop  tests  in  the  ordinary 
way  this  equivalent  length  of  loop  should  be  used,  when  the  wires 
differ,  for  L  in  the  regular  formulae. 

The  above  statement  becomes  when  expressed  in  symbols, 

°  Cb  Cb 

where  lg  and  4  are  the  lengths  respectively  of  the  good  and  faulty 
wires,  and  cg  and  cb  are  the  resistances  per  unit  length  of  the  good 
and  faulty  wires.  Le  is  the  equivalent  loop  resistance  to  use 
in  place  of  L. 

Example. 

Refer  to  (b)  par.  1205,  Fig.  1205c. 

Let  the  length  eJ  =  lg,  and  let  rg  =  cglg  be  the  resistance  of  the 
good  wire. 

Let  the  length  fJ  =  lb,  and  let  rb  =  cblb  be  the  resistance  of  the 
faulty  wire. 

Then  x  =  cbd  is  the  resistance  to  the  fault  when  d  is  the  distance 
to  the  fault. 

We  now  have 

b      rg  -\-rb  —  x      Cglg  -f  cblb  —  Cbd  /0, 

—  =   =   - ;  (Z ) 

ax  Cbd 

whence 

,  a  Cglg    ~|~    Cblb     _  d  J  /Q\ 

a  =  r- T   —    —  r— r  Lie.  (6) 

a  +  b        cb  a  +  b 

(b)  Faulty  wire  of  two  different  sizes. 

The  methods  of  par.  1207,  Eq.  (8),  and  of  par.  1208,  Eq.  (6),  will 


ART.  1210]  PRINCIPLES  OF  FAULT  LOCATION  279 

give  accurate  results  when  the  faulty  wire  is  of  two  different  sizes, 
provided  the  length  and  size  of  each  section  is  known  and  a  proper 
correction  is  applied.     This  correction  is  made  as  follows: 
Let  Fig.  1210a  represent  the  faulty  wire  of  two  sizes, 
I  =  the  length  of  the  entire  wire, 
d  =  the  length  of  one  section, 
di  =  the  length  of  the  other  section, 
s  and  si  =  the  resistance  per  unit  length  of  the  first  and 

second  sections  respectively  and 
w  =  the  total  resistance  of  the  faulty  wire. 

L +— --_ _i 

K Q, T:  >H  _  &\  *1 


r^n^izllZ-D M 

J.  Fi 

FIG.  1210a. 

We  have  two  cases  to  consider.     First,  where  the  fault  F  is  in 
the  first  section  and,  second,  where  the  fault  FI  is  in  the  second 
section. 
Let  D  =  the  distance  to  the  fault  F, 

x  =  the  resistance  to  the  fault  F, 
DI  =  the  distance  to  the  fault  FI  and 
#1  =  the  resistance  to  the  fault  FI. 

Then    in    the    first   case,   as  x  =  Bw    (where  B=  —  or  — ; 

ai        a  +  ai 

according  as  the  method  used  is  that  of  par.  1207  or  par.  1208), 

we  have 

x  =  sD  =  B(sd-{-  Sidi), 

or  D=Bsd+88ldl-  (4) 

In  the  second  case  where  FI  is  located  beyond  the  first  section  of 
1  he  wire,  we  have 

Xi  =  s  d  +  (Di  —  d)  Si  =  B  (sd  +  Si  di), 


Dl  =  B  l      +  -d.  (5) 

Si  Si 

It  will  be  noted  that  the  values  of  D  and  DI  are  not  altered  when 
s  and  Si  are  multiplied  by  the  same  constant;  hence  we  can  call 
5  and  si  ohms  per  foot,  ohms  per  meter,  or  ohms  per  1000  feet. 


280  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1210 

In  Eqs.  (4)  and  (5)  if  s  =  si}  that  is,  if  the  two  sections  are  of  the 
same  size  wire,  we  have 


=Bl  =  -l     or 


a  -f  ai 

If  the  test  shows  that  the  resistance  to  the  fault  is  equal  to  or 
less  than  the  resistance  sd  of  the  first  section  Eq.  (4)  should  be 
used,  but  if  it  is  greater  than  this  then  Eq.  (5)  should  be  used, 
(c)  Where  the  loop  consists  of  conductors  of  different  sizes* 


The  loop  may  consist  of  several  lengths  of  conductor  of  different 
sectional  areas,  as  frequently  occurs  when  cable  circuits  are  joined 
to  toll  lines.  The  distance  to  the  fault  in  this  case  may  be  calcu- 
lated by  expressing  the  lengths,  having  different  cross-sections, 
in  terms  of  what  would  be  equivalent  lengths  of  any  one  of  the 
conductors  in  the  loop.  To  do  this  the  cross-sections  and  the 
lengths  of  each  of  the  sections  in  the  loop  must  be  known.  The 
procedure  is  to  multiply  the  length  of  each  conductor  by  its  re- 
sistance per  unit  length  and  divide  the  product  by  the  resistance 
per  unit  length  of  the  conductor  of  the  size  to  which  the  others 
are  to  be  reduced.  The  following  example  will  further  explain  the 
process  : 

In  the  loop  shown  in  Fig.  121  Ob  the  conductor  in  the  cable 
section  a  to  e  consists  of  2200  feet  of  No.  19  B.  &  S.  copper  wire,  the 
conductor  in  the  section  e  to/  consists  of  1400  feet  of  No.  22  B.  &  S. 
wire  and  the  section  /  to  6,  which  may  be  considered  part  of  a  toll 
line,  consists  of  2160  feet  of  No.  12  B.  &  S.  wire.  We  shall  reduce 

*  The  description  of  this  correction  is  taken  without  material  modification 
from  the  published  literature  of  The  Leeds  and  Northrup  Company,  with  its 
permission. 


ART.  1211]  PRINCIPLES   OF  FAULT  LOCATION  281 

the  No.  19  and  the  No.  12  to  equivalent  lengths  of  the  No.  22 
wire.     Using  1000  feet  as  the  unit  of  length,  we  have 

=  1097  feet  °f  N°*  2 


is  equal  in  resistance  to  2200  feet  of  No.  19  wire. 
Also 

2160  X  1.586 


16.12 


=  212.5  feet  of  No.  22  wire,  which 


is  equal  in  resistance  to  2160  feet  of  No.  12  wire.  This  makes  the 
total  length  of  the  loop  equivalent  to  1097  +  212.5  +  1400  =  2709.5 
feet  of  No.  22  wire.  If  the  test  shows  the  fault  F  to  be  1346 
equivalent  feet  from  a,  then  1097  feet  is  in  the  section  a  to  e. 
Consequently  the  fault  must  be  1346  -  1097  =  249  feet  from  e, 
or  2200  +  249  =  2449  feet  from  a. 

(d)  Lead  Wires. 

The  testing  apparatus  cannot  always  be  brought  close  to  the 
ends  of  the  cable  and  it  then  becomes  necessary  to  use  lead  wires 
of  considerable  length  and  resistance.  In  this  case  the  simplest 
procedure  is  to  use  lead  wires  of  equal  length  and  of  the  same  size 
as  the  wires  in  the  conductor.  The  distance  to  the  fault  will  then 
be  given  from  where  the  instrument  is  located. 

If  lead  wires  cannot  be  obtained  of  the  same  size  as  the  con- 
ductors in  the  cable,  then  multiply  the  length  of  each  by  its  resist- 
ance per  unit  length  and  divide  the  product  by  the  resistance 
per  unit  length  of  the  wire  in  the  cable.  The  values  thus  found 
represent  the  equivalent  length  of  the  wire  in  the  cable  which  has 
the  same  resistance  as  each  lead  wire. 

If  occasion  arises  where  the  separate  resistance  of  each  lead 
wire  is  unknown,  it  can  be  determined  by  fastening  the  two  lead 
wires  together  at  their  far  ends  and  measuring  the  resistance  of 
the  loop.  The  point  of  junction  is  then  put  to  earth  and  the 
separate  resistances  determined  by  a  Murray-loop  test. 

121 1.  Location  of  Grounds  on  High-tension  Cables.  —  High- 
tension  power  lines  carried  on  poles  may  become  grounded  thru 
a  high-resistance  ground  at  some  one  point.  This  may  result  from 
the  breaking  of  an  insulator.  It  is  often  possible  to  locate  a 
ground  of  this  character  by  means  of  a  Murray  or  Varley-Ioop  test. 
The  power  must,  of  course,  be  taken  off  the  conductors  while  the 
test  is  being  made.  Under  these  circumstances,  however,  there 


282 


MEASURING  ELECTRICAL  RESISTANCE       [ART.  1211 


is  usually  another  line  on  the  same  poles  which  is  kept  alive  while 
the  test  is  being  made  on  the  faulty  line.  When  this  is  the  case 
the  test  is  made  difficult  from  two  causes:  Grounds  which  are 
serious  for  the  high  voltages  employed  in  the  transmission  may, 
nevertheless,  have  a  very  high  resistance,  which  makes  it  difficult 
to  pass  enough  direct  current  into  the  line  to  give  sufficient  sensi- 
bility for  the  test.  Then,  further,  the  live  wires  which  run  on  the 
same  pole  line  with  the  line  under  test  induce  in  the  latter  alter- 
nating currents  of  considerable  magnitude.  These  induced  cur- 
rents may  cause  a  serious  disturbance  of  the  galvanometer  system, 
causing  it  to  vibrate  so  violently  that  the  scale  cannot  be  read. 

Insulation  between  Cable  1  and  Gr.  Connect  A    to  C  i  and   B    to  Gr. 
••  2   "      «          «        A    to  C  2    "     B    to  Gr. 

««  land  Cable  2  Connect  A  to  C2   »nd    B    to  Cl 

G.r.  Cable 


Spark  Gap 


g      To  get  Galvanometer  Constant  disconnect 
3J       Cable  1  and  Cable  2  from  apparatus. 

Connect  A  to  Ci   and  B  to  Ci   and  short 
circuit  the  impedance  coil. 

Good 
Cable 

FIG.  1211. 

These  two  difficulties,  however,  can  generally  be  overcome.  The 
testing  galvanometer  should  be  of  the  high  sensibility  type  with 
a  suspended  coil  system,  and  the  testing  current  should  be  ob- 
tained from  a  small  generator  which  can  supply  direct  current  at 
500  volts  or  over.  The  second  difficulty  can  be  avoided  as  follows : 
There  is  joined  in  series  with  the  high  sensibility  galvanometer  a 
large  ironless  inductance  (the  larger  the  better,  but  there  is  no 
gain  by  attempting  to  increase  the  inductance  by  using  iron).  A 
cheap  paper  condenser,  of  10  or  more  microfarads,  is  then  con- 
nected as  a  shunt  around  the  inductance  and  the  galvanometer  in 
series.  This  condenser  absorbs  much  of  the  alternating  current 
and  effectually  prevents  any  serious  disturbance  of  the  galva- 
nometer system. 


ART.  1212] 


PRINCIPLES   OF  FAULT  LOCATION 


283 


A  complete  cable-testing  and  fault-locating  outfit  for  high-tension 
lines  was  designed  in  part  by  the  author  and  supplied  by  The  Leeds 
and  Northrup  Company  to  one  of  the  power  companies  at  Niagara 
Falls.  The  layout  of  circuits  employed  in  this  case  with  a  little 
study  is  self  explanatory.  It  is  given  in  Fig.  1211. 

1 2 12.  Location  of  Faults  upon  Low-tension  Power  Cables. — 
In  locating  grounds  upon  cables,  as  trolley  line  feeders,  electric 
light  and  power  cables,  etc.,  satisfactory  results  cannot  be  obtained 
by  the  Murray  or  Varley-Ioop  method  unless  special  apparatus 
of  heavy  construction  is  used.  The  resistance  of  the  contacts 
where  the  wires  are  joined  to  the  bridge  and  the  small  current 
carrying  capacity  of  a  small  diameter  slide  wire  make  accuracy 
and  sensibility  impossible  with  light  apparatus  of  ordinary 
construction. 

To  overcome  these  difficulties  a  special  bridge  for  locating  faults 
in  power  circuits  has  been  placed  upon  the  market.  In  this 


FIG.  1212. 

bridge,  which  operates  upon  the  Murray-loop  principle,  a  heavy 
manganin  slide  wire  is  used,  bent  into  circular  form.  It  is  placed 
underneath  the  top  of  the  instrument  and  a  circular  scale  and 
index  are  mounted  upon  the  top.  The  bridge  is  provided  with 
heavy  flexible  leads  about  7  feet  long.  The  terminals  of  these 
leads  end  in  clamps  of  heavy  construction  into  which  the  ends 
of  the  cable  can  be  securely  fastened.  The  contact  resistance 
is  thus  made  very  low.  Two  other  light  flexible  leads  go  from 
the  pointer  galvanometer,  which  is  self-contained  in  the  bridge 
case,  to  the  clamps  which  terminate  the  two  lead  conductors. 
The  scale,  which  otherwise  is  divided  into  1000  divisions,  has 
10  divisions  removed  from  each  end.  These  correspond  to  a 
length  of  bridge  wire  which  has  the  same  resistance  as  the  leads 
which  clamp  to  the  conductors.  The  normal  current  to  use  with 
this  bridge  is  5  amperes.  In  Fig.  1212  the  bridge  is  shown  joined 


284  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1213 

to  a  cable  of  two  different  sizes  of  wire.  Locations  can  be  made 
in  this  case  as  explained  above.  When  the  entire  loop  is  a  con- 
ductor of  one  size  the  location  is  calculated  by  the  ordinary 
Murray-loop  formula  without  modification. 

1213.  Method  of  Locating  Grounds  upon  Heavy,  Short,  Under- 
ground Cables.  —  Sometimes  very  heavy  feeder  lines,  laid  in  a 
trench,  become  grounded.  In  this  case  all  loop  methods  will  fail 
to  locate  the  ground.  Mr.  Felix  Wunsch  has  described  *  a  method 
by  which  grounds,  under  the  above  circumstances,  may  be  quite 
accurately  located. 


FIG.  1213. 

The  essential  principle  of  the  method  is  indicated  in  Fig.  1213. 
An  alternating  current  is  sent  into  the  line  at  one  end  a  and 
leaves  the  line  at  the  fault  F,  returning  to  the  generator  thru  the 
earth.  An  exploring  coil,  about  two  feet  in  diameter,  which  con- 
sists of  about  500  turns  of  No.  18  wire,  is  used.  The  terminals 
of  this  are  joined  to  a  telephone  with  a  head  band.  The  coil  is 
carried  above  ground  along  the  path  of  the  cable  until  a  point  is 
reached  where  the  sound  in  the  telephone,  caused  by  the  induction 
of  the  current  in  the  cable  upon  the  exploring  coil,  either  ceases 
or  becomes  very  greatly  diminished.  This  method  is  reported  by 
Mr.  Wunsch  to  have  given  very  good  results.  If  the  fault  is  of 
high  resistance  it  can  be  broken  down  by  applying  a  high  voltage. 
The  alternating  current  needed  is  not  much  above  one  half  am- 
pere, and  the  exploring  coil  can  be  from  10  to  15  feet  above  the 
faulty  conductor. 

A  similar  method  is  used  by  the  telephone  companies  in  locating 
faults  before  cutting  open  the  sheath  of  the  cable.  The  source  of 
current  is  a  small  induction  coil  and  cell  of  battery.  The  method 
is  extremely  accurate  and  is  readily  applied. 

1214.  The  Location  of  Opens.  —  An  aerial  wire,  or  a  con- 
ductor in  a  cable,  may  be  severed  at  some  point  so  the  circuit  is 
*  Electrical  World,  February,  1909,  vol.  xxi,  page  118. 


ART.  1214] 


PRINCIPLES  OF  FAULT   LOCATION 


285 


completely  interrupted.  This  is  called  an  open  and  it  is  important 
to  determine,  from  either  end  of  the  line,  the  distance  to  the  break; 
or  to  locate  the  "  open."  The  possibility  of  doing  this  depends 
upon  the  fact  that  every  linear  conductor  has  an  electrostatic 
capacity.  It  is  a  condenser.  The  conductor  itself  is  one  "  coat- 
ing "  or  plate  of  the  condenser;  the  dielectric  is  the  insulation 
surrounding  the  condenser,  be  this  air  or  insulating  covering 
upon  the  wire,  or  both;  the  other  coating  or  plate  of  the  condenser 
is  any  conductor  which  may  lie  parallel  with  the  other  thruout 
its  length.  In  the  case  of  an  aerial  this  would  be  the  earth  (capac- 
ity to  ground)  or  another  conductor  on  the  same  poles  (capacity 
to  a  conductor).  In  the  case  of  telephone  or  telegraph  wires  in 
a  cable  with  a  lead  sheath,  the  other  plate  of  the  condenser  would 
be  the  lead  sheath  of  the  cable  (capacity  to  ground)  or  any  one 
of  the  other  wires  in  the  cable  (capacity  to  a  conductor) . 

A  telephone  wire  in  a  cable  is  usually  twisted  with  a  return  wire 
which  constitutes  its  "  mate."  If  both  wires  of  a  pair  are  open  at 
both  ends,  then  the  two  constitute  a  condenser;  the  two  con- 
ductors, in  this  case,  being  the  condenser  plates,  and  the  double 
thickness  of  insulation  which  separates  the  conductors  being  the 
dielectric. 

Now,  the  capacity  of  a  combination  of  this  kind  is  quite  approxi- 
mately proportional  to  the  length  of  the  conductor  and,  therefore, 
if  the  capacity  of  a  conductor  (which  is  the  broken  conductor)  of 
unknown  length  is  compared  with  the  capacity  of  a  conductor  of 
known  length,  then  the  distance  to  the  open  point  is  determined. 

The  comparison  of  two  capacities  may  be  made  very  simply  with 
circuits  arranged  in  the  manner  of  a 
Wheatstone  bridge. 

In  Fig.  1214a,  r\  and  r2  are  two 
ohmic  resistances.  These  should  be 
as  free  as  possible  from  electrostatic 
capacity  or  self-induction  and  pref- 
erably should  be  fairly  high  resist- 
ances, of  the  order  of  1000  ohms. 
Ci  and  02  are  the  two  capacities  to 
be  compared.  D  is  some  form  of 
detector,  usually  a  telephone,  which 
is  responsive  to  an  alternating,  interrupted,  or  rapidly  varying 
current  of  any  kind,  This  bridge  arrangement  is  supplied  with  a 


FIG.  1214a. 


286  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1214 

current  of  this  character.  By  varying  the  ratio  of  the  two  resist- 
ances ri  and  r2  a  value  of  the  ratio  may  be  found  such  that  at  all 
times  the  potential  at  point  1  is  the  same  as  the  potential  at  point  2. 
This  equality  of  potentials  will  be  indicated  by  the  detector  D;  in 
the  case  of  a  telephone,  by  silence  in  the  telephone. 
The  condition  for  a  balance  is 

*  =  «,  (1) 

T2         Ci 

Note  that  the  capacities  are  in  reciprocal  relation  to  the  resistances. 
In  the  application  of  this  principle  to  fault  location  a  telephone 
is  invariably  used  as  the  detector.  The  source  of  variable  current 
is  a  battery  and  a  buzzer,  or  a  battery  and  a  small  hand  commu- 
tator for  quickly  reversing  the  current.  In  an  emergency  the 
current  may  be  rapidly  interrupted  by  drawing  a  metal  piece  over 
the  surface  of  a  coarse  file. 


>j&l 

FIG.  1214b. 

(a)  Good  wires  available. 

To  locate  an  open,  say,  in  a  conductor  in  a  telephone  cable  which 
contains  the  mate  of  the  broken  wire  and  another  good  pair,  the 
connections  are  made  as  in  Fig.  1214b. 

Here  the  capacities  of  the  pair  a,  b  (wire  a  broken)  and  a1}  bi 
are  indicated  by  the  hypothetical  condensers  drawn  in  dotted  line. 
ri  and  r2  are  varied  together,  or  either  of  them  alone,  until  the 
telephone  P  is  silent,  or  nearly  so. 

Then, 


As  d,  the  distance  to  the  fault,  is  proportional  to  Ci,  and  as  Z,  the 
length  of  the  good  pair,  is  proportional  to  c2, 

d-=L  (2) 


ART.  1214] 


PRINCIPLES  OF  FAULT  LOCATION 


287 


If  the  wires  a  and  61  are  aerials,  as  a  telegraph  or  electric  light 
wire,  then  p,  the  point  of  attachment  to  6  and  01,  would  be  joined 
to  the  earth  or  to  another  good  wire  on  the  same  poles,  which  'runs 
the  full  length  of,  and  is  separated  the  same  distance  from,  both 
the  good  and  the  faulty  conductor. 

For  this  location  to  be  successful  the  conductors  must  not  be 
grounded  or  crossed  and  their  far  ends  must  be  completely  open. 

(b)  When  no  good  wire  is  available. 

In  this  case.it  is  necessary  to  make  a  test,  first  at  one  end  of  the 
line  and  then  at  the  other  end.  Also  an  auxiliary  condenser  must 
be  used.  The  capacity  of  this  need  not  be  known  and  its  value 
may  be  chosen  between  wide  limits,  but  the  same  condenser  must 
be  used  thruout  the  test.  A  suitable  value  would  be  one  half 
microfarad. 


t „_ 


4==*. 


FIG.  1214c. 


Referring  to  Fig.  1214c  the  connections  are  made  first  as  in  I. 

Here  ci,  shown  in  dotted  line,  represents  the  capacity  to  ground 
of  the  open  section  a  to  o  of  the  conductor  ab.  c  is  the  capacity 
of  the  auxiliary  condenser.  For  a  balance, 


r2 


kd 


or 


(3) 


where  d  is  the  distance  from  the  end  a  to  the  open,  and  k  is  a 
constant  of  proportionality.  The  connections  are  made  next  at 
the  other  end  of  the  line  as  in  II.  Here  c%,  shown  in  dotted  lines, 
represents  the  capacity  to  ground  of  the  open  section  b  to  o. 


288  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1215 

For  a  balance,  in  this  case, 

ri_  _  c_  _         c 

r%       c%      k  (I  —  d) 

or  c  =  k  (I  —  d)  —.1  (4) 

n 

where  /  is  the  length  of  the  open  wire  ab. 
From  Eqs.  (3)  and  (4), 

J  Wl  1  fK\ 

(5) 


(c)  Another  test,  used  by  the  Bell  Telephone  Company,  which 
is  said'  to  be  extremely  useful  and  exceedingly  simple  to  apply,  is 
the  following:  A  telephone  cable  which  is  carried  into  a  building 
and  is  not  covered  with  a  lead  sheath  may  have  a  break  in  a  wire 
underneath  the  insulation.  The  exact  position  of  this  break, 
within  an  inch  or  two,  is  located  by  the  use  of  a  telephone,  a  buzzer, 
and  a  battery.  The  buzzer  has  one  terminal  put  to  earth  and  the 
other  to  the  wires  in  the  cable  at  a  free  end.  The  tester  attaches 
one  terminal  of  a  telephone  to  the  earth  and  the  other  terminal  to 
his  body.  He  then  places  his  hand  upon  the  cable  containing  the 
broken  wire.  If  he  is  on  the  side  of  the  break  to  which  the  buzzer 
is  attached  he  will  hear  a  sound  in  the  telephone.  He  moves  his 
hand  along  the  cable  and  when  he  has  passed  the  break  the  sound 
ceases.  In  this  way  the  position  of  the  break  is  narrowed  down 
and  finally  located  within  an  inch  or  two  of  its  exact  position. 
The  cause  of  the  sound  in  the  telephone  is  the  condenser  current 
which  flows  thru  the  telephone.  The  wire  of  the  cable  forms  one 
plate  of  the  condenser  and  the  tester's  hand,  which  grasps  the 
insulated  conductor,  forms  the  other  plate  of  the  condenser. 
This  test  is  very  much  used. 

1215.  Location  of  Inductive  Crosses.  —  An  inductive  cross 
(denned  in  par.  1201)  may  be  located  by  a  procedure  similar  to 
that  employed  in  locating  an  open.  It  is  necessary  in  applying 
the  test  to  have  in  the  same  cable  sheath  a  pair  of  good  con- 
ductors. The  method  requires  a  comparison  of  capacities,  when 
the  conductors  used  for  the  test  are  connected,  first  in  one  manner 
and  then  in  another.  A  setting  of  the  ratio  arms  to  give  a  bal- 
ance is  made  for  each  connection,  and  from  these  two  settings 
the  necessary  data  are  obtained  for  calculating  the  distance  to 
the  fault  or  inductive  cross.  The  method  is  carried  out  as  follows: 


ART.  1215]  PRINCIPLES  OF  FAULT   LOCATION 


289 


Connections  are  made  first  as  in  I,  Fig.  1215.     The  hypothetical 
condensers,  represented  in  dotted  line,  are  drawn  to  indicate  the 


'  1  Good 
•2  Pair 


~l^.-c 


'd 


-¥-  i^<-*- 

/    \ I 4 


JG 


>\ 


II 

FIG.  1215.    . 

capacity  per  unit  length  of  conductor  between  the  pair  of  good 
wires  1,  2;  a  good  wire  and  its  mate  3,  4;  and  this  same  good 
wire  and  the  wire  4,  mated  beyond  the  fault  with  wire  6  of  another 
pair. 

A  balance  is  obtained  with  these  connections,  by  varying  the 
resistances  r\  and  r2.     Connections  are  then  made  as  in  II,  Fig. 
1215,  the  dotted  lines  indicating  an   equally  good  alternative 
arrangement.     The  hypothetical  condensers  drawn  in  dotted  line 
show  the  capacities  per  unit  length  of  conductor  as  they  would  be- 
come with  these  connections.     A  balance  is  again  obtained,  the 
ratio  arms  taking  the   values  r/  and  r2'.     The  distance  to  the 
fault,  by  an  approximate  formula,  is  now  calculated  as  follows: 
Let  I   =  the  length  of  the  cable, 
d  =  the  distance  to  the  fault, 
c  =  the  capacity  per  unit  length  of  a  conductor  and  its 

mate  and 

GI  =  the  capacity  per  unit  length  of  a  conductor  and  a  con- 
ductor of  another  pair. 


290                MEASURING  ELECTRICAL  RESISTANCE  [ART.  1216 

In  case  I, 

713  —            c^  n\ 

T_  ~  ~ J    i    ..   n        j\'  \*) 


In  case  II, 

***  d  /^\ 

T77— ~^'  (2) 


For  brevity  let 

T  V    ' 

—  =  a.     and    -^  =  6. 
ri  ri' 

Then,  from  Eq.  (1), 

C^Cai-d)'  (3) 

and  from  Eq.  (2) 

_  c  [Z  -  b  (I  -  d)] 
Cl~  bd 

Hence, 

I  —  a  d    _  I  —  b  (I  —  d)  ,  . 

a(l-d)  ~        ~bdT 
From  Eq.  (5)  we  find 

d  =  b®a  _  1\  _.  a  l-  (6) 

If  in  Eq  (6)  we  replace  a  by  its  value  —  and  6  by  its  value  — , , 

we  obtain  for  the  distance  to  the  fault, 

f   f         >\ 
7  _         ^2  (/2  —  ri )         1  .  . 

r2/(2r2-r1)-r2r/t< 

Eq.  (7)  is  not  rigidly  true,  because  all  of  the  capacity  relations 
between  different  conductors  were  not  taken  into  account.  It  is, 
however,  sufficiently  exact  for  practical  purposes. 

1216.  Comments  on  Practice  and  Accuracy  in  Fault  Location. 
—  Tho  the  principles  of  fault  location  and  the  formulae  used  are 
relatively  quite  simple,  difficulties  are  apt  to  arise  in  their  appli- 
cation in  the  field.  The  chief  cause  of  these  difficulties  is  that 
conditions,  which  are  assumed  to  be  constant  in  deducing  the 
formulae,  prove  variable  in  practice.  Thus  in  settled  districts  no 
two  widely  separated  points  upon  the  surface  of  the  earth  are  at 
exactly  the  same  potential.  For  this  reason,  if  a  conductor  makes 
contact  with  the  earth  at  two  points,  stray  currents  will  flow  in  the 
line.  Then  also  the  proximity  of  other  lines  carrying  currents 
which  alternate  or  large  direct  currents  which  vary  will  often 
induce  stray  currents  in  the  testing  circuit.  These  may  cause 


ART.  1216]  PRINCIPLES  OF  FAULT  LOCATION  291 

erratic  movements  of  the  galvanometer  which  seriously  interfere 
with  the  measurement. 

The  resistance  of  a  fault,  especially  a  ground,  may  vary  greatly 
while  the  test  is  in  progress.  A  ground  may  be  due  to  a  moist 
condition  of  the  insulation  which  the  testing  current  dries  out,  and 
the  ground  will  disappear  while  the  test  is  in  progress.  This  is 
known  as  a  disappearing  ground. 

Then  another  serious  cause  of  trouble,  which  may  become  very 
puzzling  and  exasperating,  is  a  bad  contact  resistance  at  some 
unknown  point  in  the  loop  circuit.  If  this  contact  resistance  is 
constant  false  results  will  be  obtained,  but  if,  as  often  happens,  the 
bad  contact  is  variable,  it  becomes  impossible  to  obtain  a  balance 
while  the  cause  of  the  difficulty  is  misjudged. 

Again  two  faults  may  be  present  on  a  conductor.  The  location 
then  will  only  give  some  intermediate  point  between  the  faults. 
If  the  resistance  of  one  of  the  two  faults  is  steady  while  that  of 
the  other  varies,  the  point  of  balance  on  the  bridge  will  shift  in  a 
puzzling  way.  The  cause  of  this  would  be  difficult  to  distinguish 
from  the  effects  of  a  poor  contact.  The  best  procedure,  when 
there  are  two  faults,  is  to  cut  the  wire  between  the  faults  and  locate 
each  one  separately.  The  existence  of  two  faults  may  be  dis- 
proved by  testing  from  each  end  of  the  line.  If  both  locations 
place  the  fault  at  the  same  point  there  is  only  one. 

Any  method  which  gives  only  the  resistance  to  the  fault  is  in- 
ferior to  one  which  gives  the  distance  to  the  fault  as  a  fraction  of 
the  total  length  of  the  cable.  In  calculating  distances  from  resist- 
ances it  should  be  remembered  that  copper  wire  varies  in  resist- 
ance about  0.4  of  1  per  cent  per  degree  C.,  and  the  temperature  of  a 
long  conductor  may  vary  considerably  from  one  point  to  another. 
Then  also  a  small  variation  from  standard  gauge  in  the  conductor 
may  mislead  one  in  calculating  the  distance  from  the  resistance. 

Experience  shows  that  copper  wire  in  telephone  cables  laid  under 
ground  will  run  about  10  per  cent  higher  in  resistance  in  summer 
than  in  winter  in  the  State  of  Pennsylvania. 

The  most  important  error,  however,  is  likely  to  arise  from  the 
fact  that  the  conductors  are  usually  longer  than  the  cable  sheath, 
since  pairs  of  conductors  are  twisted  together  in  telephone  cables. 
Even  aerials  will  be  longer,  due  to  the  sag  of  the  wire,  than  the 
distance  measured  along  the  pole  line. 

It  will  be  noted  that  in  the  methods  which  have  been  given,  the 


292  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1216 

first  two  excepted,  the  resistance  of  a  fault,  a  cross  or  a  ground, 
does  not  enter  into  any  of  the  measurements.  Also  that  the  gal- 
vanometer is  so  placed  that  neither  the  potential  differences  existing 
in  the  earth,  nor  any  electromotive  force  at  the  fault  itself,  can  send 
a  current  thru  the  galvanometer.  Many  methods  which  might 
be  given  for  locating  crosses  or  grounds  have  not  been  mentioned 
because  they  involve  measuring  the  resistance  of  the  fault  itself 
or  expose  the  galvanometer  to  possible  earth  currents  or  electro- 
motive forces.  Such  methods,  some  of  which  are  well  known  in 
connection  with  fault  locations  upon  marine  cables,  work  well  with 
artificial  lines  in  the  laboratory,  but  they  give  uncertain  and 
unsatisfactory  results  when  used  upon  land  lines  in  the  field.  For 
this  reason  we  have  omitted  giving  them,  but  the  interested  student 
will  find  the  standard  methods  of  this  character  fully  explained  in 
Kempe's  "  Hand  Book  of  Electrical  Testing,"  and  in  other  works 
upon  marine  cable  testing. 

If  a  helper  has  been  instructed  to  make  a  connection  at  the  far 
end,  it  is  possible  for  the  tester  to  prove  or  disprove  that  he  has 
done  so.  One  method  of  ascertaining  whether  a  connection  has 
been  made  at  the  far  end,  to  a  wire  which  can  be  connected  to  the 
testing  set,  is  carried  out  as  follows :  Prepare  for  a  test  of  the  elec- 
trostatic capacity  of  the  conductor  in  question  by  the  deflection 
method.  Take  a  deflection  before  and  after  the  supposed  con- 
nection has  been  made.  If  upon  closing  the  circuit,  the  latter  de- 
flection is  the  larger  the  connection  has  been  made. 

Another  and  preferable  method  is  to  join  the  two  wires, 
which  are  to  be  connected  by  the  helper  at  the  far  end,  to  the 
X  posts  of  the  testing  set.  The  switches  of  the  set  are  arranged 
for  making  a  loop  resistance  measurement.  A  resistance  is  un- 
plugged in  the  rheostat  which  is  greater  than  the  resistance  of  the 
loop  can  possibly  be.  Before  the  two  wires  are  joined  at  the  far 
end  the  pointer  of  the  galvanometer  will  deflect  to  one  end  of  the 
scale  corresponding  to  infinite  resistance  for  X.  As  soon  as  the 
helper  makes  the  connection  the  pointer  will  deflect  to  the  opposite 
end  of  the  scale  corresponding  to  a  resistance  less  than  that  un- 
plugged in  the  rheostat. 

In  giving  the  precision  with  which  a  fault  is  located  it  is  cus- 
tomary to  give  not  the  relative,  or  per  cent,  value  but  the  absolute 
precision  expressed  in  feet  or  meters.  In  measurements  of  this 
character  the  important  matter  is  the  actual  distance  in  feet  or 


ART.  1217]  PRINCIPLES  OF  FAULT  LOCATION  293 

meters  that  the  location  is  out,  regardless  of  the  length  of  line. 
It  is  well,  however,  to  state  this  latter  as  giving  additional  infor- 
mation regarding  the  circumstances  under  which  the  location  was 
made. 

1217.   A  Word  on  Fault-locating  Apparatus.  —  Fault  locations 
upon  land  lines  can  be  made,  if  necessary,  with  comparatively 


FIG.  1217a. 

simple  apparatus  which  a  skillful  tester  can  devise  and  assemble 
from  material  usually  to  be  found  about  an  electric  station. 
However,  in  connection  with  the  work  of  a  large  telephone 
equipment,  the  locating  of  faults  is  an  important  and  frequent 
operation.  It  is  economy,  therefore,  to  use  fault-locating  apparatus 
devised  especially  for  portability,  speed  and  precision. 

It  is  impossible  to  give  the  space  here  required  to  describe,  even 
in  outline,  the  many  different  forms  of  fault-locating  and  cable- 
testing  apparatus  which  instrument  makers,  here  and  abroad, 
have  placed  upon  the  market.  We  shall  merely  mention  two 
which  have  had  an  extensive  use  in  this  country.  One  is  the  well- 
known  portable  cable-testing  set  designed  by  Henry  W.  Fisher. 
With  this  set,  which  is  robust  and  complete  in  every  respect,  the 
following  tests  are  readily  effected: 

Location  of  crosses  and  grounds. 

Location  of  breaks  or  opens  in  cables. 

Conductor  resistance  measurements. 

Liquid  resistance  measurements. 

Insulation  resistance  measurements. 

Capacity  measurements. 


294 


MEASURING  ELECTRICAL  RESISTANCE       [ART.  1217 


The  outside  appearance  of  this  set  is  shown  in  Fig.  121 7a.  The 
other  apparatus  referred  to,  with  the  design  of  which  the  author 
was  largely  connected,  is  the  "  Lineman's  Fault  Finder."  This  is 
shown  in  Fig.  1217b. 


FIG.  12175. 

The  essential  feature  of  the  apparatus  is  a  uniform  resistance, 
which  lies  in  a  circle  and  is  about  100  ohms.  By  a  special  con- 
struction, it  is  arranged  so  that  contact  can  be  made  at  any  point 
along  it,  and  it  is  therefore  equivalent  to  a  very  high  resistance 
slide  wire.  It  has  a  moving  contact  and  a  uniform  scale  of  1000 
divisions.  In  series  with  this,  there  are  two  resistances,  which 
may  be  short-circuited  by  switches.  One  has  exactly  the  same 
resistance  as  the  wire.  There  is  a  resistance  of  100  ohms,  and  it 
is  the  fixed  resistance  of  the  bridge  arrangement  for  resistance 
measurements.  Resistances  of  1000  ohms  and  9000  ohms  are 
connected  to  the  battery  post  to  protect  the  battery  and  the 
apparatus  from  excessive  current.  The  9000  ohms  may  be  short- 
circuited  by  a  switch,  Other  features  are  a  self-contained  battery 


ART.  1217]  PRINCIPLES  OF  FAULT  LOCATION  295 

and  a  galvanometer  of  the  type  described  in  par.  1501,  and  three 
switches  which  permit  the  connections  to  be  quickly  and  unmis- 
takably made  for  the  following  uses: 

Measurement  of  conductor  resistances. 

Murray  and  Varley-loop  tests,  and,  when  a  telephone  and  buzzer 
are  used  as  accessories,  the  location  of  opens. 

Both  of  the  above  sets  are  manufactured  by  The  Leeds  and 
Northrup  Company,  of  Philadelphia,  Pa. 


CHAPTER  XIII. 

MEASUREMENT  OF  TEMPERATURE  BY  THE 
MEASUREMENT  OF  RESISTANCE.* 

1300.  Remarks  on  Temperature  and  Thermometry.  —  The 

measurement  of  a  physical  quantity  implies,  generally,  the  numeri- 
cal comparison  of  the  quantity  with  a  certain  selected  quantity 
of  the  same  kind  taken  as  a  unit.  Temperature,  however,  can- 
not be  treated  as  a  quantity  in  the  same  sense.  It  is  rather  to  be 
considered  as  a  state  in  which  matter  is  found,  and  all  temperature 
measurements  are  made  by  comparing  the  changes  in  some  form 
of  matter  produced  by  heat.  As  shown  by  Lord  Kelvin  as  early 
as  1848,  temperature  may  be  expressed  on  a  scale  which  is  inde- 
pendent of  any  particular  form  of  matter,  but  this  thermodynamic 
scale  cannot  be  used  in  actual  temperature  measurements,  which, 
in  practice,  consist  in  comparing  the  change  in  some  particular 
form  of  matter  produced  by  changes  in  temperature. 

Certain  gases  change  in  volume  under  constant  pressure  or 
change  in  pressure  under  constant  volume  in  a  nearly  regular 
manner  with  equal  increments  of  temperature,  as  estimated  on 
the  thermodynamic  scale.  Gas  thermometers  have,  therefore, 
naturally  been  chosen  as  standards  with  which  to  compare  the 
changes  in  various  forms  of  matter,  which  changes  may  then  serve 
as  a  convenient  means  of  temperature  measurement.  The  present 
upward  range  of  the  gas  thermometer  scale  is  1550°  C.,  with  a 
probable  error  of  plus  or  minus  2°  C.,f  and  the  melting-point  of 
pure  platinum  is  known  within  plus  or  minus  5°  C.  and  is  assigned 
the  value  1755°  C.  The  melting-points  of  many  other  metals  are 
known  with  varying  degrees  of  accuracy,!  and  these  melting-points 
of  the  metals  constitute  fixed  temperatures  which  may  be  used 
for  the  calibration  of  various  temperature-measuring  devices. 

The  science  of  thermometry,  especially  its  extension  into  high- 
temperature  pyrometry,  is  far  too  extensive  to  be  even  touched 

*  Portions  of  this  chapter  are  taken  from  an  article  by  the  author  in  the 
Proc.  of  the  A.  I.E. E.,  1906. 

f  Dr.  A.  L.  Day,  Trans,  of  the  Faraday  Soc.,  Nov.,  1911,  pages  142  and  144. 

296 


ART.  1301]          MEASUREMENT  OF  TEMPERATURE  297 

upon  here,  and  its  consideration  does  not  belong  to  a  work  of  this 
kind,  but  the  resistance  thermometer,  which  is  one  of  the  best 
devices  for  the  measurement  of  temperature,  may  with  propriety 
be  briefly  described  as  well  as  the  methods  employed  for  deter- 
mining temperature  by  its  use. 

1301.  Electrical-resistance  Thermometry.*  —  Electrical  resist- 
ance thermometry  is  possible  because  very  many  electrical  con- 
ductors change  in  resistance  with  change  of  temperature  in  a 
perfectly  definite  manner. 

The  percentage  change  in  resistance  of  the  pure  metals  with 
temperature  is  larger  than  that  in  the  volume  of  gases,  and  over 
twenty  times  as  great  as  the  volume  change  in  mercury.  Thus, 
the  coefficient  of  expansion  of  nitrogen  gas  is  0.00367  +  ,  and  of 
mercury  0.00018  -f,  while  the  coefficient  of  increase  of  resistance 
of  pure  nickel  is  about  0.0041  per  degree  C.  between  0°  and  100°  C. 

A  change  in  electrical  resistance  can  be  measured  with  greater 
ease  and  far  greater  precision  than  a  change  in  volume  of  a  liquid 
or  a  gas.  A  change,  in  either  a  high  or  a  low  electrical  resistance, 
can  be  measured  when  it  is  one  part  in  a  hundred  thousand.  Thus, 
the  sensitiveness  of  the  electrical-resistance  method  of  measuring 
temperature  is  very  great.  In  the  use  of  the  bolometer,  where  the 
electrical-resistance  method  of  measuring  temperature  is  carried 
to  its  greatest  sensitiveness,  temperature  changes  as  small  as  one 
ten-millionth  of  a  degree  C.  are  said  to  be  detectable. 

For  the  electrical-resistance  method  of  measuring  temperature 
to  be  of  utility  the  resistance  which  is  measured  must  always  return 
to  the  same  value  when  brought  back  to  the  same  temperature. 
Fortunately,  experience  has  shown  that  when  the  proper  resistance 
materials  are  chosen,  and  due  precautions  in  their  treatment  have 
been  used,  the  reliability  of  the  method  in  this  respect  is  very 
satisfactory.  A  properly  constructed  resistance  thermometer,  if 
not  exposed  to  too  high  a  temperature,  will  maintain  its  calibration 
better  and  longer  than  the  best  mercury  thermometer,  which  is 
usually  subject  to  small  alterations  and  irregularities  due  to  elastic 
after-effects  in  the  glass. 

*  A  valuable  treatment  of  this  subject  may  be  found  in  the  Bulletin 
of  the  Bureau  of  Standards,  Vol.  6,  Nov.,  1909,  page  149.  Article  by  C.  W. 
Waidner  and  G.  K.  Burgess.  A  bibliography  of  the  subject  is  given  there, 
pages  223-230.  See  also,  "Measurement  of  High  Temperatures,"  Burgess 
and  Le  Chatelier,  1912  edition. 


298  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1301 

As  the  pure  metals  are  greatly  elevated  in  temperature,  the 
rate  of  increase  in  resistance  with  temperature  generally  changes. 
Thus,  over  extensive  temperature  ranges  there  are  no  metals  of 
which  the  resistance  is  even  approximately  a  linear  function  of 
temperature.  Small  impurities  in  the  pure  metals  affect  also  the 
amount  as  well  as  the  law  of  their  change. 

These  facts  make  it  unlikely  that  an  electrical-resistance  tem- 
perature scale  will  be  found  bearing  such  definite  relations  to  the 
absolute-temperature  scale  that  it  will  serve  conveniently  for  a 
standard  scale  of  reference  in  the  same  manner  as  does  the  scale  of 
the  gas  thermometer.  When,  however,  the  means  are  available,  it 
is  relatively  easy  to  determine  experimentally  the  relation  between 
the  electrical  resistance  of  any  particular  specimen  of  wire  and  the 
temperature  for  a  working  range  of  the  gas  thermometer  of  900° 
or  1000°  C.  An  electrical-resistance  thermometer  can  then  be 
made  of  this  specimen  of  wire,  and  it  will  serve  as  a  standard  with 
which  other  resistance  thermometers  may  be  very  simply  and 
easily  compared. 

The  law  of  variation  of  electrical  resistance  with  temperature 
in  the  case  of  platinum  has  been  investigated  by  Callendar  and 
Griffiths,  and  several  others.  It  has  been  shown  that  in  the 
case  of  platinum  the  following  relation  exists  between  the.  tem- 
perature t,  as  measured  on  the  air  thermometer,  and  the  resistance 
of  platinum: 

Let  pt  be  a  so-called  "  platinum  temperature  "  as  defined  by  the 
relation  r>  r> 

'  ~ 


.tiioo  —  -fi/o 

where  R0  is  the  resistance  of  a  given  specimen  of  platinum  at  0°, 
Rioo  at  100°,  and  Rt  at  t°,  all  measured  on  the  centigrade  scale. 
It  has  been  shown  that  placing 


t  —  pt   =  5  \    —  :r7rp:  +      r^ 


expresses  the  difference  between  the  "  platinum  temperature  " 
and  the  temperature  as  measured  on  the  air  thermometer.  This 
"  difference  formula,"  as  it  is  called,  holds  to  within  0.1°  C.  up  to 
500°  C.  and  within  0.5°  C.  up  to  1000°  C.  In  this  formula  5  is  a 
coefficient  which  varies  with  the  particular  specimen  of  platinum 
used.  For  very  pure  platinum  it  is  about  1.5,  and  larger  for  im- 
pure specimens.  To  determine  5  the  resistance  of  the  thermometer 


ART.  1301] 


MEASUREMENT   OF  TEMPERATURE 


299 


is  measured  at  the  three  known  temperatures,  0°  C.,  100°  C.,  and 
444.6°  C.,  the  boiling-point  of  sulphur.  The  authors  referred  to 
give  convenient  methods  of  using  the  difference  formula  to  con- 
vert the  temperatures  as  given  by  the  platinum-resistance  tem- 
perature scale  to  degrees  centigrade  as  given  on  the  scale  of  the  air 
thermometer. 

In  the  relation  (1)  above  the  quantity  RiOQ  —  RQ  is  called  Fi} 
the  fundamental  interval.  It  is  a  constant  quantity  for  any  par- 
ticular thermometer. 

F- 
The  quantity  C  =    nn  *     is  called  the  fundamental  coefficient. 


100  R< 


As 
we  have 


pt  =  100 


t  —  RO 


Cpt  = 


Rt- 


(3) 


Thus,  the  purer  the  platinum  the  greater  will  be  the  coefficient  C. 
As  examples  of  the  values  of  the  above  constants  we  give  the  fol- 
lowing data  taken  from  tests*  made  by  the  National  Bureau  of 
Standards  upon  two  platinum-resistance  thermometers,  called  A 


Thermometer  A 

Thermometer  B 

#0  =  21.3476 
Fi=  4.4067 
C=  0.00206426 
5=    1.571 
Diameter  of  wire  =  0.01  cm. 

Ro=  3.  48779 
^=1.34298 
0  =  0.00385052 
5  =  1.504 
Diameter  of  wire  =  0 

.015cm. 

and  B.  The  current  thru  the  thermometers  in  the  above  test 
was  0.004  ampere  and  0.010  ampere  respectively. 

If  we  plot  resistance  as  ordinates  and  gas  thermometer  degrees 
'as  abscissae,  the  curve  obtained  for  platinum  is  always  slightly 
concave  toward  the  axis  of  X  and  is  parabolic  in  form.f 

When  pure  nickel  is  used  for  resistance  thermometers,  the  re- 
sistance variation  obeys  another  law.  Prof.  C.  F.  Marvin  has 

*  Bulletin  of  the  Bureau  of  Standards,  Vol.  6,  page  156,  1909. 

t  For  a  more  extended  discussion  of  formula  (2)  and  for  a  description 
of  methods  for  reducing  platinum  temperatures  to  the  gas  scale,  consult 
"Measurement  of  High  Temperatures,"  Burgess  and  Le  Chatelier,  1912  edi- 
tion, Chapter  V. 


300  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1301 

shown  *  that  the  nickel-resistance  curve  is  very  closely  represented 
by  the  equation 

Loge  R  =  a  +  ml,  (4) 

where  R  is  resistance  of  thermometer  in  ohms,  t  the  temperature 
in  degrees  C.,  and  m  and  a  are  constants.  In  some  particular  cases 
this  equation  became 

LogeR  =  1.0854  +  0.001699 1, 
and  again, 

Loge  R  =  1.9004  +  0.001818 1, 
and  again, 

LogeR  =  0.9614  +  0.001450*. 

These  equations  were  tested  in  the  range  —25°  C.  to  75°  C.  with 
an  error  never  greater  than  0.1°  C.  and  again  in  the  range  0°  C. 
to  375°  C.  with  an  error  not  exceeding  0.9°  C. 

The  simple  meaning  of  the  above  relation  [Eq.  (4)]  is  that  pure 
nickel  wire  increases  in  resistance  by  the  same  per  cent  of  its  re- 
sistance at  the  beginning  of  an  increment  of  temperature  for  every 
equal  increment  in  temperature  anywhere  in  the  range  —  25°  C. 
to  350°  C. 

The  law  is  sufficiently  accurate  to  be  relied  upon  for  work  not 
requiring  a  precision  greater  than  1°  C.  over  the  range  mentioned 
above,  and  for  short  ranges  of  50°  C.,  or  less,  reliance  may  be 
placed  upon  the  law  to  0.1°  C.,  or  better. 

As  far  as  known,  no  other  metal  obeys  the  law  of  nickel.  Be- 
cause of  this  law  for  nickel  the  temperature-resistance  curve  may 
be  located  by  observing  the  resistance  at  only  two  temperatures, 
say  0°  C.  and  100°  C.  When  resistance  is  plotted  as  ordinates 
and  temperature  as  abscissae  the  curve  will  always  be  convex 
toward  the  axis  of  X.  It  follows  that,  if  a  certain  length  of  nickel 
wire  is  joined  in  series  with  a  certain  length  of  platinum  wire,  a 
combination  resistance  thermometer  may  be  made  which,  over 
short  ranges,  will  have  a  change  with  temperature  which  is  prac- 
tically linear. 

To  engineers  and  those  who  make  industrial  uses  of  resistance 
thermometers  the  theoretical  side  of  the  subject  is  of  minor 
interest.  There  is  a  practical  procedure  which  may  be  adopted 
that  makes  it  unnecessary  for  manufacturers  or  users  to  give 
consideration  to  these  methods  of  standardization  of  resistance 
*  Physical  Review,  April,  1910,  pages  522-528. 


ART.  1301]          MEASUREMENT  OF  TEMPERATURE  301 

thermometers.  The  instrument  maker  may  carefully  construct  a 
resistance  thermometer  to  serve  as  a  standard  and  send  this  from 
time  to  time  to  the  National  Bureau  of  Standards  at  Washington. 
The  Bureau  will  measure  the  resistance  of  this  thermometer 
over  a  wide  range,  at  several  known  temperatures  given  by 
their  standard  resistance  thermometers,  and  furnish  a  certificate 
giving  the  relations  which  are  found  between  temperature  and 
resistance  of  the  thermometer  submitted  for  calibration.  The 
instrument  maker  may  then  use  this  thermometer  as  a  standard 
with  which  other  thermometers  are  easily  calibrated.  This  is 
done  by  direct  comparison  in  an  oil  bath  for  medium  temperatures, 
and  in  a  specially  constructed  electric  furnace  for  high  tempera- 
tures. Cold  brine,  or  liquid  air,  or  other  means  may  be  used  for 
making  the  comparison  at  low  temperatures. 

The  feature  of  paramount  importance  in  the  use  of  electrical 
resistance  thermometers  is  the  constancy  with  which  they  main- 
tain their  calibration.  This  subject  has  received  considerable 
attention,  especially  in  the  case  of  thermometers  made  of  platinum 
wire,  and  the  results  observed  have  proved  the  entire  reliability 
of  this  material  for  temperatures  not  exceeding  1000°  C.  It  is 
highly  probable  that  other  materials  will  behave  in  an  entirely 
regular  manner  if  not  subjected  to  too  high  temperatures. 

Careful  investigations  of  the  constancy  of  other  materials 
than  platinum  that  are  suitable  for  resistance  thermometers  are 
needed.  But  the  investigations  so  far  made  show  that  where 
permanent  alterations  in  resistance  occur  these  may  usually  be 
traced  to  causes  which  proper  precautions  may  avoid.  Thus,  the 
material  selected  for  the  thermometer  may  be  by  nature  of  an 
unstable  character.  Iron,  for  example,  is  an  unsuitable  metal  to 
use.  The  material  may  contain  impurities  which  by  vaporization, 
crystallization,  or  otherwise,  cause  the  resistance  to  alter  gradually. 
The  wire  of  which  the  thermometer  is  made  may  have  been  sub- 
jected to  mechanical  strains  which  gradually  work  out  with  re- 
peated heatings,  thus  altering  the  resistance.  If  the  material  is 
one  which  does  not  oxidize,  it  may  still  be  greatly  affected  at  high 
temperatures  by  absorbing  gaseous  impurities.  Thus,  a  nickel-wire 
or  a  platinum-wire  thermometer  heated  to  400°  C.  in  a  brass  tube 
is  ruined  by  absorbing  the  metallic  vapors  given  off.  For  the 
same  reason  all  metal  solderings  near  the  resistance  wire  are  liable 
at  high  temperatures  to  give  off  vapors  which  affect  the  permanent 


302  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1302 

resistance,  besides  making  liable  the  formation  of  local  resistance 
at  the  joints. 

Proper  construction  and  choice  of  materials  can  remove  all  the 
above  causes  of  permanent  alterations.  It  may  be  that  in  the 
case  of  platinum,  to  some  extent  at  least,  and  more  so  in  other 
materials,  slow  permanent  alterations  in  resistance  occur,  the 
cause  of  which  is  not  known.  Only  extended  investigations  can 
give  the  limits  of  these  possible  alterations.  Enough  work  has 
been  done,  however,  to  show  that  for  even  very  refined  work  the 
reliability  of  platinum  and  some  other  materials  is  sufficient  if 
temperatures  too  high  are  not  exceeded. 

In  resistance  thermometry  practical  details  of  construction  are 
all  important.  The  chief  of  these  will  now  be  considered. 

1302.  Construction  of  Resistance  Thermometers.  —  The  best 
material  of  which  to  construct  a  resistance  thermometer  depends 
upon  the  temperature  range  to  be  measured,  as  well  as  upon  the 
physical  qualities  of  the  available  materials. 

Constancy  of  composition  and  other  practical  considerations 
seem  to  limit  the  choice  to  a  few  of  the  pure  metals,  usually  in  the 
form  of  wire.  The  metal  which  has  received  the  most  study  is 
platinum.  It  can  be  used  over  a  very  wide  temperature  range, 
and  can  be  obtained  under  the  name  of  Herseus  platinum  in  a  state 
of  great  purity.  This  material  answers  every  requirement  of  re- 
sistance thermometry,  except  that  it  is  very  costly.  A  substitute 
for  platinum  should,  therefore,  be  sought  and  used  wherever  it 
will  serve  as  well.  This  substitute  should  be  inexpensive  and 
obtainable  in  a  pure  state.  It  is  desirable  that  it  should  have  a 
high  specific  resistance,  combined  with  a  large  temperature  coeffi- 
cient. It  should  be  unoxidizable  under  usable  conditions,  and 
withstand  a  high  temperature  without  deterioration  or  permanent 
alteration  in  resistance.- 

An  examination  of  the  pure  metals  shows  that  these  conditions 
are  best  met  by  nickel.  The  author  has  had  many  thermometers 
constructed  of  this  wire  for  temperatures  ranging  from  —  40°  C. 
to  300°  C.,  and  has  found  it  reliable  in  this  range.  It  has  a  higher 
coefficient  than  the  purest  platinum,  that  of  nickel  being  about 
0.0041  per  degree  between  0°  C.  and  100°  C.,  pure  platinum  being 
0.0039,  and  commercial  platinum  but  about  0.002.  The  specific 
resistance  of  pure  nickel  and  pure  platinum  is  in  the  ratio  of  about 
933  to  1000. 


ART.  1302]          MEASUREMENT  OF  TEMPERATURE  303 

It  may  here  be  remarked  that  a  determination  of  the  tempera- 
ture coefficient  of  the  metallic  elements  offers  usually  a  very  deli- 
cate test  of  their  purity,  and  specimens  of  nickel  and  platinum 
which  show  a  low  temperature  coefficient  can  positively  be  con- 
sidered as  impure  and  inferior  for  use  in  resistance  thermometers. 

Another  test  of  interest,  especially  on  wires  intended  for  use 
in  thermocouples,  is  to  attach  the  two  ends  of  a  short  length  to 
the  terminals  of  a  very  sensitive  galvanometer,  and  to  pass  a 
flame  along  the  wire.  If  the  galvanometer  gives  positive  and 
negative  deflections  of  considerable  magnitude,  the  wire  may  be 
known  to  be  unhomogeneous,  and  liable  to  have  parasitic  currents 
set  up  in  it  when  exposed  to  high  temperatures.  A  pure  nickel 
and  a  pure  platinum  wire  should  show  little  of  this  effect. 

The  particular  purpose  for  which  a  resistance  thermometer  is 
to  Be  used  largely  determines  its  special  features  of  construction. 
Broadly  classified,  resistance  thermometers  are  particularly  useful 
in  the  following  cases: 

1.  Measurement  of  all  temperatures  below  —  40°  C.,  the  freez- 
ing point  of  mercury. 

2.  Measurement  of  all  temperatures  up  to  1000°  C.,  when  the 
temperature  is  to  be  taken  at  a  place  where  it  cannot  be  directly 
observed. 

3.  Measurement  of  temperatures  below  1000°  C.,  and  above 
the  range  of  the  mercury  thermometer. 

4.  Measurement  of  all  temperatures  below   1000°  C.,   which 
must  be  photographically  or  otherwise  recorded. 

5.  Determinations  of  small  temperature  differences  or  varia- 
tions  for  which  the  mercury  thermometer   is    not    sufficiently 
sensitive. 

It  is  evident  from  the  above  classification  that  there  can  be  no 
general  form  or  type  of  construction  of  a  resistance  thermometer. 
Each  special  requirement  must  be  met  by  the  instrument  maker, 
who  should  be  guided  in  his  designs  by  experience  and  a  study  of 
the  conditions.  The  form  of  thermometer  having  been  chosen, 
the  particular  method  of  reading  the  resistance  variations  and  of 
expressing  them  in  degrees  should  have  particular  care,  for  in 
nearly  every  case  which  arises  different  requirements  must  be  met. 

Resistance  thermometers  for  use  below  140°  C.  are  of  relatively 
simple  construction,  for  in  this  case  silk-insulated  nickel  wire  may 
be  used.  Certain  precautions,  nevertheless,  need  attention.  The 


304  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1302 

mass  of  the  wire  used  and  that  of  the  body  on  which  it  is  wound 
should  be  small,  or  the  temperature  of  the  resistance  wire  will  lag 
behind  any  changing  temperatures  which  are  being  measured, 
and  lead  to  erroneous  indications.  The  wire  must  be  so  chosen 
in  respect  to  size  and  resistance  that  the  heating  of  the  wire  by  the 
measuring  current  shall  be  negligible. 

The  constancy  of  any  wound  resistance  depends  largely  upon 
the  treatment  to  which  it  is  subjected  after  being  wound.  The 
winding  of  the  wire  introduces  strains,  which  gradually  work  out, 
causing  variations  in  the  permanent  resistance.  This  certain 
result  is  avoided  by  an  artificial  "  aging,"  which  consists  in  main- 
taining the  wire  for  several  hours  or  days,  before  the  thermometer 
is  calibrated,  at  a  temperature  higher  than  that  at  which  it  will 
be  used. 

Insulating  Rod  holding  Mica  Winding  Form. 

Platinum  Winding  on  Mica  Form. 
,  Metal  Case. 


Lead  Wires  in  Grooves 
in  Insulating  Rod. 


FIG.  1302a. 


It  is  needful  to  finish  the  terminals  of  the  wire,  especially  if 
short,  in  such  a  manner  that  no  local  variations  in  resistance  can 
occur  at  the  joints.  As  a  rule  the  terminals  should  be  hard  silver 
soldered  for  low-temperature  thermometers,  and  for  high-tem- 
perature thermometers  all  joints  exposed  to  the  high  temperature 
must  be  welded  joints. 

Generally,  the  resistance  wire  should  be  protected  by  a  casing. 
When,  however,  as  in  the  measurement  of  moderate  temperatures 
of  gases  or  insulating  fluids,  the  wire  can  be  directly  in  contact 
with  whatever  is  to  have  its  temperature  determined,  the  resist- 
ance thermometer  assumes  the  surrounding  temperature  very 
quickly,  far  surpassing  the  mercury  thermometer  in  this  respect. 
If  a  casing  must  be  used,  it  should  be  so  shaped  that  the  ratio  of 
its  surface  to  its  volume  is  large,  and  the  construction  should  aim 
to  reduce  to  a  minimum  the  heat  which  is  conducted  along  the 
case  or  which  is  distributed  by  air  convection  within  it.  Repro- 
ductions are  here  given  of  two  types  of  resistance  thermometers 
designed  for  the  measurement  of  low  or  moderate  temperatures. 

The  thermometer  shown  in  Fig.  1302a  was  constructed  for  use 


ART.  1302]          MEASUREMENT  OF  TEMPERATURE  305 

in  measuring  and  recording  with  great  precision  the  temperature 
differences  between  two  brine  mains.  The  average  temperature 
of  the  brine  was  about  —  37°  C.  and  the  average  difference  of 
temperature  between  the  two  mains  was  about  1.5°  C.  The 
allowable  error  was  0.01°  C.,  and  hence  great  care  in  the  construc- 
tion of  the  thermometers,  as  well  as  in  the  rest  of  the  apparatus, 
was  required.  This  thermometer  was  wound  with  No.  35  platinum 
wire,  of  great  purity.  Its  resistance  at  room  temperature  was 
about  80  ohms.  It  is  probable  that  nickel  wire  would  have  served 
as  well,  but  because  of  the  better  known  properties  of  platinum 
and  the  importance  of  the  experiment  platinum  was  selected. 

No.32  B.  &  S.  Nickel  Wire.     (Silk  Insulation) 
/  Silk  Insulation 


" 

FIG.  1302b. 

It  should  be  noted  that  the  steel  case  is  long  and  small  in 
diameter,  that  the  winding  ends  well  below  the  nut  which  screws 
into  the  brine  main,  and  that  the  wire  is  wound  on  a  light  frame 
of  mica,  having  a  minimum  of  mass.  A  small  sudden  change 
in  the  temperature  of  the  brine  was  followed  by  the  thermometer 
to  within  about  0.005°  C.  within  two  minutes. 

Fig.  1302b  is  a  sectional  view  of  a  form  of  resistance  thermometer 
made  for  the  purpose  of  measuring  the  temperature  of  the  soil 
at  different  depths  where  the  thermometers  are  permanently 
buried.  The  winding  is  in  the  form  of  a  skein,  and  No.  32  silk- 
insulated  nickel  wire  is  used.  To  insure  permanency  the  wire 
should  be  kept  immersed,  after  winding,  in  hot  paraffin  for  three 
or  four  days.  The  changes  in  the  temperature  of  the  soil  are  very 
slow,  and  hence  there  is  no  need  to  provide  against  a  temperature 
lag  of  the  thermometer  winding. 

The  resistance  is  made  large,  about  100  ohms  at  20°  C.,  and 
the  winding  is  encased  in  a  brass  tube  filled  with  paraffin.  The 
lead-covered  leads  are  soldered  with  a  wiped  joint  to  the  brass 
tube,  thus  preventing  the  entrance  of  moisture,  which  has  to  be 
carefully  avoided.  The  resistance  of  the  thermometer  being  high, 
the  change  in  the  resistance  of  the  leads  is  entirely  negligible. 

A  somewhat  similar  construction  would  be  suitable  for  measur- 


306 


MEASURING  ELECTRICAL  RESISTANCE      [ART.  1302 


ing  the  temperature  of  the  interior  of  stored  material,  such  as 
grain,  tobacco,  hay,  wheat,  etc.,  also  for  measuring  the  temperature 
of  cold-storage  rooms.  Any  number  of  such  thermometers  can  be 
located  at  different  places  and  be  connected  by  a  switch,  one  at  a 
time,  to  a  single  reading  device  which  reads  directly  in  degrees 
Fahrenheit  or  Centigrade.  The  methods  of  reading  these  and 
other  resistance  thermometers  will  be  presently  described. 


Porcelain. 
Tube    Platinum         Tube 
Lead  Wire 


Platinum 
Wi'nding 


FIG.  1302c. 


II 


Resistance  thermometers  give  often  the  most  accurate  and  con- 
venient means  of  measuring  high  temperatures  up  to  1000°  C.  or 
possibly  more.  It  is  stated  by  Le  Chatelier*  that  experiments 
carried  out  at  the  National  Physical  Laboratory,  England,  showed 
that  throughout  the  temperature  range  of  1000°  C.  the  agreement 
between  the  scales  of  the  platinum-resistance  and  the  thermo- 
electric pyrometers  tested  was  within  0.5°  C. 

Such  statements  as  the  above,  however,  are  true  only  when 
the  resistance  thermometers  have  been  constructed  in  a  particular 
manner  to  avoid  alterations  and  deteriorations  in  the  wire  that 
are  sure  to  result  at  high  temperatures  with  improper  construction. 
Platinum  heated  red  hot  and  exposed  to  certain  gases,  as  hydrogen 
or  metallic  fumes,  absorbs  impurities  which  permanently  alter  its 
resistance  and  often  render  it  extremely  brittle. 

Accumulated  experience  has  shown  that  for  temperatures  above 
a  red  heat  (525°  C.  to  600°  C.  for  all  materials)  the  design  of  the 
thermometer  should  embody  the  general  features  shown  in  the 
illustration,  Fig.  1302c,  I  and  II. 

In  the  thermometer  here  illustrated,  the  winding  is  a  pure 
*  "High  Temperature  Measurements,"  page  105,  1904  edition. 


ART.  1302]  MEASUREMENT  OF  TEMPERATURE  307 

Harseus  wire,  its  purity  being  shown  by  its  temperature  coefficient, 
which  is  about  0.0039°  at  100°  C.  This  wire,  No.  35  B.  &  S.,  is 
wound  bare,  on  a  frame  of  thin  mica,  in  such  a  manner  as  to  touch 
only  the  edges  of  the  mica.  The  winding  is  36  turns  to  the  inch. 
The  mica  frame  is  made  by  matching  together  at  right  angles  two 
pieces  of  mica  sheet,  of  the  shape  shown  in  Fig.  1302d. 


Wire 
I 


G 


C 


Mica  Sheet 


FIG.  1302d. 

As  the  winding  touches  only  at  the  edges  of  the  mica,  only  a 
small  percentage  of  its  length  can  become  contaminated  by  any 
possible  action  of  a  solid  material.  The  lead  wires,  by  a  method 
of  compensation  to  be  later  described,  do  not  enter  into  the  resist- 
ance which  is  measured,  and  may  be  of  a  less  pure  platinum 
than  the  resistance  winding.  These  lead  wires  are  either  three 
or  four  in  number,  according  to  the  method  of  compensation 
adopted.  They  are  insulated  from  each  other  by  being  passed 
through  tubes  of  porcelain. 

For  temperatures  above  the  fusion  point  of  hard  glass,  porcelain 
tubes  especially  constructed  for  this  work  by  the  Royal  Berlin 
Porcelain  Works  are  the  most  satisfactory  material  for  a  casing. 

The  interior  parts  of  the  thermometer  shown  in  Fig.  1302c,  II, 
are  designed  to  be  easily  withdrawn  from  the  tube  for  examination, 
and  again  replaced.  In  the  particular  case  shown,  the  winding 
was  made  13  cms  long,  to  give  the  thermometer  a  high  resistance. 
This  is  generally  an  advantage,  where  the  conditions  permit,  as 
the  contact  resistances  in  the  measuring  device  are  then  small  in 
comparison,  and  greater  sensibility  is  more  easily  obtained. 

It  was  legitimate  to  make  the  winding  long  for  the  case  shown, 
as  this  thermometer  was  designed  to  measure  the  temperature  of 
hot  gases  which  would  surround  the  porcelain  tube  more  than,  half 
way  to  its  head.  If,  however,  the  temperature  of  the  place  to  be 
measured  is  uniform  over  a  small  space  only,  then  the  winding 
should  be  as  short  and  as  much  concentrated  at  the  end  of  the 
tube  as  possible,  and  so  permit  of  placing  the  entire  winding  in 
the  hot  place,  the  temperature  of  which  is  to  be  measured. 


308  MEASURING  ELECTRICAL  RESISTANCE      [ART.  1303 

Thermocouples  have  in  this  respect  an  advantage  over  a  resist- 
ance thermometer  as  above  designed,  for  the  end  of  the  thermo- 
couple is  a  very  small  body,  that  may  be  closely  located  at  the 
place,  where  the  temperature  is  to  be  observed.  This  consideration 
led  the  author  to  design  another  form  of  resistance  thermometer 
which  will  be  shown  to  combine  the  advantages  of  both.  A  de- 
scription of  this  is  best  given,  however,  under  methods  of  reading 
resistance  thermometers,  which  we  shall  now  consider. 

1303.  Methods  of  Reading 'Resistance  Thermometers.  —  As 
previously  stated,  the  National  Bureau  of  Standards  at  Washington 
will  furnish  the  instrument  maker  with  a  certificate  giving  the  con- 
nection between  the  electrical  resistance  and  the  temperature  of  a 
selected  standard  resistance  thermometer,  and  the  calibration  of 
other  thermometers  is  reduced  to  comparing  their  resistances  with 
that  of  the  standard  when  all  are  brought  to  equal  temperatures, 
In  the  case  of  high  temperatures,  a  specially  constructed  electric 
furnace  is  used  for  the  purpose.     The  problem,  then,  of  reading 
temperatures  with  thermometers  thus  calibrated  resolves  itself  into 
measuring  their  resistance  in  a  simple  manner  when  subjected  to 
different  temperatures. 

The  resistance  being  known,  the  temperature  may  be  taken 
from  a  previously  plotted  curve,  or  the  resistance-measuring 
device  may  be  constructed  to  read  directly  in  degrees  Centigrade 
or  Fahrenheit.  The  convenience,  simplicity,  precision,  and  re- 
liability with  which  these  measurements  can  be  made  largely 
determine  the  practical  and  commercial  usefulness  of  resistance 
thermometers.  The  continuous  recording  of  temperatures  given  by 
resistance  thermometers  is  another,  but  closely  related,  problem, 
but  one  which  cannot  here  receive  our  attention. 

The  available  and  useful  methods  of  determining  resistances  for 
measuring  temperatures  may  be  classified  as  follows : 

Slide- wire  bridge  method. 

Differential  galvanometer  method. 

By  resistance-thermometer  bridge  with  two  traveling  contacts. 

By  use  of  dial  bridges. 

Kelvin-double-bridge  method  of  reading  temperatures. 

Direct-deflection  method  of  reading  temperatures. 

1304.  Slide-wire  Bridge  Method.  —  This  is  a  very  convenient 
zero  method  to  employ,  especially  when  the  reading  instrument 
has  a  scale  calibrated  to  read  directly  in  degrees.     The  slide-wire 


ART.  1304] 


MEASUREMENT  OF  TEMPERATURE 


309 


bridge  may  have  its  connections  arranged  in  either  of  two  useful 
ways.  The  first  is  less  precise,  but  more  convenient.  The  connec- 
tions are  given  diagrammatically  in  Fig.  1304a. 


Ga 


FIG.  1304a. 

jPi,  Tz,  T3,  etc.,  represent  any  number  of  resistance  thermom- 
eters; y,  y  are  the  thermometer  leads,  which  should  be  alike  but 
may  be  of  any  length.  Contact  can  be  made  with  any  ther- 
mometer by  means  of  a  simple  sliding  switch  S.  The  resistances  r, 
ri,  r2,  should  be  about  equal  to  each  other  and  to  the  resistance  of 
the  thermometer  when  at  a  mean  temperature.  The  resistance  of 
the  slide  wire  I  should  be  such  as  will  take  care  of  only  the  variation 
in  resistance  of  the  thermometers. 

In  an  actual  construction,  the  contact  p  would  move  over  a 
circularly  disposed  wire  and  scale.  This  scale  may  be  divided 
into  arbitrary  divisions,  and  reference  be  made  to  a  curve,  to 
obtain  the  temperature  of  any  thermometer  corresponding  to  a 
given  setting  for  a  balance.  In  this  case,  the  different  thermom- 
eters need  to  be  made  of  only  approximately  the  same  resistance. 
The  scale  may,  however,  without  great  difficulty,  be  graduated 
to  read  directly  in  degrees  when  used  with  a  thermometer  of  a 
particular  resistance  and  temperature  coefficient. 

If,  however,  many  thermometers  are  to  be  read  on  the  same 
scale,  they  must  be  adjusted  to  exact  equality  both  in  respect  to 
resistance  and  temperature  coefficient.  This  last  adjustment  can 


310 


MEASURING  ELECTRICAL  RESISTANCE       [ART.  1304 


be  made  by  using  a  certain  resistance  of  manganin  in  series  with 
those  thermometers  which  have  too  high  a  coefficient. 

The  arrangement  of  connections  shown  does  not  entirely  com- 
pensate for  changes  in  the  resistance  of  the  leads.  The  error, 
however,  would  not  exceed  from  this  cause  0.1°  C.  in  an  ordinary 
case.  The  obvious  advantage  of  making  the  connections  in  this 
way  is  that  while  nearly  complete  compensation  is  obtained,  each 
thermometer  has  only  two  lead  wires  and  one  common  terminal 
connecting  all  the  thermometers  to  the  galvanometer.  The  man- 
ner of  making  the  bridge  connections  according  to  the  second 
arrangement  is  shown  in  Fig.  1304b. 


FIG.  1304b. 

By  connecting  the  bridge  in  this  manner  and  choosing  the  ratio 
arms  equal,  the  resistance  of  the  leads  y,  y,  entirely  eliminate. 
Thus,  the  value  of  any  resistance  Xi,  X^  etc.,  is 

X  =  R-(l-2a)=a  constant  +  2  a. 

This  method,  while  perfectly  compensating,  requires  that  two 
pairs  of  leads  shall  be  carried  to  each  thermometer.  This  is  a 
decided  disadvantage  where  many  thermometers  are  to  be  read 
at  a  distance  on  one  bridge.  The  method  recommends  itself  when 
the  highest  possible  precision  is  required.  In  this  method  also 
the  scale  may  be  calibrated  in  degrees,  if  desired. 

The  balance  point  on  the  wire  in  either  of  the  above  methods 
may  be  found  with  a  telephone,  but  preferably  with  a  galvanometer. 


ART.  1304] 


MEASUREMENT  OF  TEMPERATURE 


311 


A  pointer  galvanometer  of  portable  type,  such  as  that  described 
in  par.  1501,  is  amply  sensitive  for  the  purpose. 

The  illustration,  Fig.  1304c,  shows  a  completed  instrument, 
designed  for  portability. 


FIG.  1304c. 

A  temperature  measurement  is  made  by  slightly  depressing  the 
button,  which  closes  the  battery  circuit,  and  then  rotating  the 
pointer  until  the  galvanometer  shows  a  balance.  The  position  of 
the  pointer  then  gives  on  the  scale  beneath  it  the  temperature  in 
degrees  Fahrenheit  or  Centigrade.  The  same  kind  of  instrument 
is  equally  well  adapted  to  reading  low  or  high  temperatures. 

One  mechanical  feature  of  this  instrument  deserves  mention  as 
being  of  extreme  usefulness.  This  is  the  construction  of  the  slide 
wire  for  use  with  high-resistance  thermometers.  If  made  in  the 
customary  way,  which  is  to  use  a  single  fine  wire,  it  would  have 
to  be  very  small  in  diameter  and  delicate  in  order  to  secure  the 
necessary  resistance  in  the  length  that  can  be  used.  Thus,  a 
100-ohm  nickel  thermometer  would  change  about  40  ohms  with 


312 


MEASURING  ELECTRICAL  RESISTANCE      [ART.  1305 


100°  C.  This  objection  is  entirely  overcome  by  winding  in  a 
spiral  an  insulated  wire  for  the  slide  wire,  as  described  in  par.  401. 
By  the  use  of  such  a  spiral,  some  30  times  the  resistance  of  a  single 
wire  may  be  obtained.  It  gives  extremely  fine  variations  in  re- 
sistance, as  the  slider  moves  over  it,  and  makes  a  contrivance 
that  is  mechanically  substantial  and  not  liable  to  wear  or  become 
loose. 

1305.  Differential-galvanometer  Method.  —  Another  arrange- 
ment for  reading  the  resistance  of  resistance  thermometers  is  that 
in  which  a  differential  galvanometer  is  used  in  the  manner  already 
very  fully  described  in  pars.  301  and  302.     The  author  is  inclined 
to  believe  that,  all  things  considered,  this  differential  galvanom- 
eter arrangement  offers  more  advantages  than  any  other.     The 
scale  and  mechanical  parts  would  not  differ  essentially  from  those 
shown  in  Fig.  1304c. 

1306.  Resistance  Thermometer  Bridge  with  Two  Traveling 
Contacts.  —  It  is  possible,  however,  to  gain  the  advantage  of 


w 


FIG.  1306. 

using  only  three  wires  leading  to  the  thermometer  and  have  the 
lead  wires  entirely  compensate  and  yet  use  a  Wheatstone-bridge 
arrangement  with  a  galvanometer  of  ordinary  type  which  is  not 
differential.  The  way  in  which  this  can  be  done  without  intro- 
ducing contact  resistances  in  the  measuring  circuit  was  first  pro- 
posed by  Mr.  Morris  E.  Leeds.  It  is  as  follows: 

Represent  a  Wheatstone-bridge  network  as  in  Fig.  1306.  Here 
T  represents  the  resistance  of  a  resistance  thermometer  and  y,  y 
represent  the  equal  resistances  of  two  lead  wires  to  the  thermom- 
eter. Ga  and  Ba  are  respectively  a  galvanometer  and  a  battery. 
b,  a,  c,  and  d  are  resistances.  Let  p\  and  p^  represent  two  contact 


ART.  1306]  MEASUREMENT  OF  TEMPERATURE  313 

points  which  can  be  moved  to  different  points  upon  the  resistances 
included  between  points  1  and  2  and  between  points  2  and  5. 
Let  b  always  be  the  resistance  between  point  3  and  pi,  a  the 
resistance  between  pi  and  pz,  and  c  the  resistance  between  pz 
and  point  5. 

Then,  for  any  positions  of  pi  and  p2)  we  have,  by  the  ordinary 
law  of  the  Wheatstone  bridge, 


b  +  y      d  +  y  +  T 
from  which  we  derive 


(1) 

y  (c  -  o)  /ON 


From  Eq.  (2)  it  appears  that,  if  matters  are  so  arranged  that  the 
resistance  a  is  always  maintained  equal  to  the  resistance  c,  the 
term  y  (c  —  a)  will  disappear,  and  we  shall  have 

T-b-d,  (3) 

where  d  is  constant  and  may  be  made  zero. 

This  result  may  be  accomplished  by  a  mechanical  arrangement 
which  will  cause  the  contact  points  pi  and  pz  to  always  move 
together.  Assume  that  the  bridge  is  balanced  when  pi  is  at  point 
1  and  pz  is  at  point  2  and  that  in  this  case  the  resistance  between 
Pi  and  p2  is  ai  and  equal  to  the  resistance  Ci  between  p2  and  point 
5.  Now  let  T  increase  so  that  in  order  to  maintain  a  balance  of 
the  bridge  b  must  be  increased  or  pi  must  be  moved  towards  the 
point  2  by  a  distance  6ai  and  at  the  same  time  let  the  contact 
Pz  move  toward  the  point  5  by  a  distance  5ci.  Then  the  resist- 
ance included  between  pi  and  pz  is  now  di  —  5ai  +  dci  and  the 
resistance  included  between  pz  and  point  5  is  Ci  —  5ci.  But  to 
make  the  term  y  (c  —  a)  disappear  we  must  so  move  pz  that 

Ci  —  dci  =  ai  —  dai  +  5ci.  (4) 

If,  originally,  we  have  made  c\  =  ai}  we  obtain 

(5) 


Eq.  (5)  shows  that  if  the  resistance  of  the  slide  wire  1,  2,  5  is 
uniform  thruout  then  pi  must  be  moved  twice  as  far  as  pz.  In 
practice  the  resistance  of  the  slide  wire  from  point  1  to  point  2 


314 


MEASURING  ELECTRICAL  RESISTANCE      [ART.  1307 


would  be  given  twice  the  resistance  per  unit  length  of  the  slide 
wire  from  point  2  to  point  5,  and  thus  the  actual  distances 
thru  which  pi  and  p2  would  have  to  move  would  be  the  same. 
With  this  arranged  the  arms  which  carry  the  contacts  pi  and  pz 
may  be  rigidly  fastened  together,  and  the  value  of  T  will  always 
be  given  by  the  difference  of  the  resistances  b  and  d  regardless  of 
the  value  of  the  lead  resistances  y,  y,  provided  these  are  alike. 

1307.  Use  of  Dial  Bridges  for  Temperature  Measurements.  — 
The  slide-wire  bridge  directly  calibrated  in  degrees  is  a  very  useful 
and  rapid  reading  device,  but  for  precision  work,  combined  with 
robustness  of  construction,  some  easily  read  form  of  dial  or  plug 
Wheatstone  bridge  may  be  more  useful. 

When  the  connections  are  made  as  shown  in  Fig.  1307a,  and  the 
resistances  n  and  r2  are  equal,  the  resistances  of  the  leads  y}  y 
eliminate.  This  requires  that  the  total 
resistance  in  the  rheostat  shall  equal  the 
resistance  of  the  thermometer,  which  for 
this  reason,  as  well  as  for  sensibility,  etc., 
should  be  high,  and  that  the  brush  or  plug 
contacts  used  should  be  well  made  and  of 
negligible  resistance.  Since  no  resistance 
varies  with  temperature  in  a  strictly  linear 
way,  a  dial  or  plug  bridge  cannot  be 
calibrated  to  read  directly  in  degrees  and 
also  accurately.  The  studs  or  plug  holes 
should  be  numbered  decimally,  and,  from 
the  setting  obtained  for  a  balance,  the 
temperature  is  gotten  by  referring  to  a 
curve.  Thus,  each  thermometer  in  an  in- 
stallation has  its  own  curve,  and  new  thermometers  may  be  added 
without  reference  to  the  old.  This  method  is  very  convenient  for 
an  installation  of  a  large  number  of  thermometers,  because  of  the 
small  number  of  wires  that  have  to  be  installed. 

Fig.  1307b  shows  four  thermometers  of  a  large  installation  used 
for  measuring  the  temperature  of  gases  up  to  650°  C.  in  the  plant 
of  a  large  chemical  company  manufacturing  sulphuric  acid.  Here 
it  was  desired  that  ignorant  workmen  should  take  the  temperatures 
without  gaining  information  as  to  what  they  were.  The  dial 
settings  for  obtaining  a  balance  were  reported  to  the  office,  where 
a  clerk  looked  up  the  corresponding  temperatures  on  the  curves. 


ART.  1308]  MEASUREMENT  OF  TEMPERATURE 


315 


The  thermometers  used  in  these  installations  were  of  the  form 
illustrated  in  Fig.  1302c. 


T4 


as 


FIG.  1307b. 

1308.  Kelvin  Double-bridge  Method  of  Reading  Tempera- 
ture. —  The  resistance  thermometer  as  designed  for  high-tem- 
perature work,  if  wound  to  a  suitable  resistance,  is  necessarily  of 
considerable  size.  This  unfits  it,  as  compared  with  thermo- 
couples, for  taking  the  temperatures  of  small  places  or  at  points. 
Moreover,  the  thermometers,  besides  requiring  considerable  skill 
to  construct,  are  costly  and  more  or  less  fragile.  These  disad- 


316  MEASURING  ELECTRICAL  RESISTANCE      [ART.  1308 

vantages  are  largely  overcome  by  a  method  developed  by  the 
author  which  v/ill  now  be  described. 

The  principle  made  use  of  is  to  measure  the  changing  resistance 
of  low-resistance  platinum  thermometers  by  means  of  the  Kelvin 
double  bridge  already  fully  discussed  in  par.  609. 

With  these  bridge  connections  a  resistance  thermometer  of 
0.01  ohm  can  be  measured  with  precision.  By  taking  advantage 
of  this  bridge  as  a  reading  device,  a  high-temperature  thermometer 
can  be  constructed  as  shown  in  Fig.  1308a. 

Heraeus  Platinum1 

jrcolaiu  Tube 

Potential  Lead  Wire 
19— 


vLava  Current  Lead  Wire, 

Insulating       Heraeus  Platinum 
Mica\  Tube  not  necessary. 

Tip  to  solder  Wire  to" 

FIG.  1308a. 

The  small  spiral  of  resistance  wire  to  be  measured  shown  at  the 
end  of  the  porcelain  tube  is  of  No.  20  Herseus  platinum.  The 
current  and  potential  leads  are  of  a  cheaper  grade  of  platinum.  In 
fact,  it  is  a  positive  advantage  to  have  the  potential  leads  of  an 
impure  platinum,  because  of  its  low  temperature  coefficient,  which 
may  be  about  0.6  that  of  pure  platinum.  The  connections  as 
arranged  for  measuring  a  number  of  thermometers  would  be  as 
shown  in  Fig.  1308b. 

To  measure  a  temperature  with  this  arrangement,  the  terminals 
p,  p'  are  moved  by  a  switch  to  the  potential  terminals  of  the 
thermometers  to  be  measured,  while  the  thermometers  to  the 
right  of  the  one  being  measured  are  cut  out  of  circuit  by  y,  which 
keeps  the  resistance  of  the  "  yoke  "  low  as  required  by  theory. 
A  balance  on  the  galvanometer  is  obtained  by  moving  the  plug  N 
and  the  slider  S.  The  slide  wire  on  which  S  moves  would  consist 
of  a  substantial  manganin  wire  lying  over  a  scale,  marked  off  in 
degrees  Centigrade,  if  it  is  desired  to  make  the  bridge  direct- 
reading.  The  only  uncertain  element  in  the  method  is  the  possi- 
bility of  the  ratio  T  and  TJ  becoming  variable  in  an  unknown  way 
through  a  change  in  the  resistance  of  that  portion  of  the  potential 


ART.  1309]          MEASUREMENT  OF  TEMPERATURE 


317 


leads  which  lie  in  the  thermometer  tube.  This  uncertainty,  how- 
ever, is  practically  avoided  if  the  resistance  a  is  made  sufficiently 
high.  Calculation  shows  that  if  a  is  chosen  as  high  as  250  ohms, 
the  maximum  error  from  this  cause,  with  a  thermometer  con- 
structed like  that  shown  in  Fig.  1308a,  will  not  exceed  0.1°  C. 
The  resistance  a  may,  however,  be  as  high  as  1000  or  even  5000 
ohms,  thus  practically  reducing  the  error  to  zero. 


T4 


^Extension  Coils  for  Bridge  Wire 

FIG.  1308b. 


The  necessity  of  having  high  resistance  in  the  ratio  coils  re- 
quires that  the  galvanometer  used  shall  have  a  greater  sensibility 
than  can  easily  be  gotten  in  a  portable  pointer  instrument.  There 
are,  however,  several  very  convenient  forms  of  semi-portable 
suspended-coil  types  of  galvanometers,  having  an  attached  tele- 
scope and  scale  which  are  amply  sensitive  for  the  purpose. 

1309.  Direct-deflection  Method  of  Reading  Temperatures. — 
Direct-deflection  methods  of  measuring  physical  quantities,  as 
well  as  temperatures,  depending  as  they  do  upon  the  calibration 
of  a  scale,  seldom  have  the  precision  of  zero  methods.  They 
possess,  however,  the  advantage  of  showing  more  clearly  the 
variations  as  they  occur  in  changing  quantities,  while  no  manipu- 
lative action  is  required  on  the  part  of  the  observer.  For  these 
reasons,  largely,  all  commercial  electrical-measuring  instruments 


318 


MEASURING  ELECTRICAL  RESISTANCE       [ART.  1309 


are  of  the  deflection  type,  though  inferior  in  precision  to  the  null 
methods  used  for  calibration  and  other  purposes  in  the  laboratory. 
A  principle  may  be  employed,  however,  by  which  the  con- 
venience and  rapidity  of  the  deflection  method  is  in  part  combined 
with  the  precision  of  the  null  method.  This  principle  consists  in 
setting  with  dials  or  plugs  so  a  balance  with  the  quantity  being 
measured  is  nearly  obtained.  The  deflection  of  a  calibrated  de- 
flection instrument  then  gives  the  additional  small  quantity  which 
must  be  subtracted  or  added,  to  obtain  the  exact  value  of  the 
quantity  measured,  in  this  case  a  temperature. 


FIG.  1309. 

The  realization  of  this  principle  is  embodied  in  various  com- 
mercial instruments,  one  of  which  is  illustrated  in  Fig.  1309. 

The  instrument  operates  upon  the  principle  shown  by  the  con- 
nections given  in  Fig.  1307a.  The  deflection  indicator  is  of  the 
same  construction  as  a  switchboard  voltmeter.  When  the  plug  is 
in  any  one  block  of  the  row  of  blocks  the  temperature  indicated 
is  that  marked  upon  the  block  plus  the  reading  of  the  deflection 


AKT.  1310]  MEASUREMENT  OF  TEMPERATURE  319 

instrument.  The  range  of  the  scale  is  100°  C.,  so  if  the  plug  were 
in  the  block  stamped  500  and  the  pointer  stood  at  57  the  tem- 
perature would  be  557°  C.,  or,  if  the  plug  were  in  the  next  block 
to  the  right,  the  reading  would  be  600°  plus  the  scale  reading,  and 
so  on.  Instruments  of  this  kind  can  be  given  various  ranges  up 
to  1000°  C.  and  are  very  convenient  for  the  use  of  workmen  in 
manufacturing  establishments.  They  should  be  operated  upon  a 
fairly  constant  voltage. 

1310.  Deflection  Methods;  Using  Constant  Currents.  —  It 
is  often  required  to  read  and  to  record  temperature  differences, 
possibly  very  small  differences,  which  must  be  determined  with 
high  accuracy.  A  thermo-couple  is  customarily  employed  for 
this  purpose,  placing  one  junction  in  the  place  of  higher  and  one 
in  the  place  of  lower  temperature.  Resistance  thermometers 
may,  however,  be  employed  to  advantage,  especially  if  the  tem- 
perature difference  is  small  and  great  precision  is  required. 

The  author  has  been  connected  with  the  design,  calibration  and 
use  of  a  temperature-difference  recording  apparatus  which  em- 
bodied the  highest  refinements  in  this  kind  of  measurement, 
commercially  applied.  A  brief  description  of  this  apparatus  will 
best  explain  the  methods  to  employ,  the  precautions  that  should 
be  used  to  obtain  precision,  and  the  results  which  may  be  obtained 
in  this  class  of  work. 

The  requirement  was  to  obtain  a  continuous  photographic 
record  of  the  temperature  difference  between  two  brine  mains 
carrying  brine  for  refrigeration  purposes.  The  temperature  of 
the  brine  in  one  main  was  about  —38°  C.  and  in  the  other  about 
—36.5°  C.  It  was  sought  to  have  the  error  at  all  times  not 
greater  than  0.01°  C.  By  taking  a  photographic  trace  of  the 
temperature  difference  at  each  instant,  and  by  obtaining  with  a 
planimeter  the  average  height  of  the  ordinates  expressing  tem- 
perature differences,  the  average  temperature  difference  for  any, 
period  of  time  could  be  found.  This  result  was  secured. 

An  ordinate  500  mm.  high  corresponded  to  5°  C.  The  deflec- 
tion instrument  used  was  a  D'Arsonval  galvanometer  of  the  mirror 
type,  of  special  construction.  The  record  was  traced  on  photo- 
graphic paper,  known  to  the  trade  as  "  rotograph  "  paper.  This 
paper  was  wound  on  a  brass  cylinder  about  55  cms.  long  and 
12.5  cms.  in  diameter.  By  means  of  a  clock  movement  the  cyl- 
inder made  one  revolution  in  12  hours.  Two  cylinders  were 


320  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1310 

provided,  so  that  an  exposed  one  could  be  immediately  replaced 
by  an  unexposed  cylinder. 

The  galvanometer  was  placed  in  one  end  of  a  box  about  1.2  m 
long.  By  suitable  optical  devices,  the  spot  of  light,  about  1  mm 
in  diameter,  was  reflected  from  the  galvanometer  mirror  upon 
the  slowly  rotating  cylinder  covered  with  the  sensitive  paper. 
The  movements  of  the  spot  of  light  were  parallel  to  the  axis  of 
the  cylinder  and  proportional  to  the  temperature  difference. 
The  source  of  light  was  an  incandescent  lamp.  Another  optical 
device  cast  another  spot  of  light  upon  a  translucent  scale,  where 
the  deflections  could  at  any  time  be  observed. 

Two  platinum-resistance  thermometers,  exactly  alike,  were 
used,  one  being  placed  in  each  brine  main  at  a  distance  of  several 
hundred  feet  from  the  recorder,  being  connected  with  it  by  lead- 
covered  compensated  leads.  One  of  these  thermometers  is  de- 
scribed in  connection  with  Fig.  1302a.  Each  thermometer  with 
its  leads  formed  an  arm  of  a  Wheatstone  bridge.  The  two  other 
arms  were  made  of  equal  manganin  resistances,  each  200  ohms. 
When  the  thermometers  were  at  the  same  temperature,  the  bridge 
was  balanced,  and  the  galvanometer  deflection  read  zero;  that  is, 
the  spot  of  light  fell  on  the  scale  at  the  same  point  as  it  would  with 
the  circuit  open.  A  fixed  mirror  reflected  a  spot  of  light  which 
made  a  trace  near  the  center  of  the  paper,  which  served  as  a 
reference  line  from  which  the  extent  of  the  deflections  could  be 
measured.  When  the  two  thermometers  were  at  different  tem- 
peratures, the  deflections  were  very  nearly  proportional  to  the 
difference  in  temperature  between  them,  whatever  might  be  the 
mean  value  of  the  two  temperatures,  but  depended  upon  the  cur- 
rent entering  the  bridge.  Hence,  to  maintain  the  empirical  cali- 
bration of  the  scale,  it  was  necessary  to  provide  that  the  current 
thru  the  bridge  should  remain  constant  to  within  the  percentage 
precision  at  which  the  apparatus  was  designed  to  operate.  The 
manner  of  doing  this,  as  well  as  the  general  plan  of  the  method, 
is  best  explained  by  referring  to  Fig.  1310a. 

The  source  of  current  was  a  battery  of  eight  storage  cells.  This 
current  could  be  held  constant  within  a  fraction  of  one  per  cent  by 
occasionally  varying  the  rheostat  resistance  in  the  battery  circuit. 
The  current  was  known  to  have  the  standard  value  when  the 
galvanometer  G  in  the  standard-cell  circuit  gave  no  deflection. 
It  was  found  in  practice  that  the  rheostat  resistance  had  to  be 


ART.  1310]          MEASUREMENT  OF  TEMPERATURE 


321 


changed  only  a  few  times  in  a  day,  and  then  only  by  small  amounts. 
Since  the  scale  was  calibrated  so  that  1  mm.  was  equal  to  0.01°  C., 
the  galvanometer  had  to  be  fairly  sensitive.  Only  90  ohms,  R  in 
the  diagram,  could  be  used  in  its  circuit.  The  galvanometer  coil 
had  333  ohms  of  copper-wire  winding,  and  since  copper  changes 
about  0.4  of  1  per  cent  in  resistance  per  degree  C.,  it  was  neces- 


FIG.  1310a. 

sary  to  avoid  changes  in  the  galvanometer  sensibility  due  to  room 
temperature  changes.  The  temperature  of  the  box  enclosing  the 
galvanometer  was,  therefore,  held  constant  within  about  0.5  of  1 
degree  by  means  of  an  electric  thermostat. 

Since  it  is  impossible  to  adjust  two  resistance  thermometers 
to  exact  equality,  when  at  low  temperatures,  the  difficulty  was 
avoided  by  shunting  each  thermometer,  one  with  10,000  ohms, 
and  the  other  with  a  resistance  near  10,000  ohms,  which  thus 
made  both  thermometers  act  in  balancing  the  bridge  as  if  they 
were  exactly  equal  when  at  the  same  temperature. 

In  calibrating  the  apparatus,  a  necessary  adjustment  was  made 


322 


MEASURING  ELECTRICAL  RESISTANCE       [ART.  1311 


by  placing  both  thermometers  in  a  tank  containing  well-stirred 
brine  at  about  —  35°  C.,  and  then  varying  one  of  the  shunts  Si 
and  S2,  until  the  galvanometer  showed  no  deflection.  Another 
adjustment  was  made  by  placing  one  thermometer  in  one  brine 
tank  and  the  other  in  another  brine  tank.  The  temperature 
difference  between  these  brine  tanks  could  be  controlled,  and  this 
difference  was  accurately  measured  by  taking  a  great  many 
readings  with  specially  constructed  mercury  thermometers  with 
Reichenstalt  certificates.  The  corresponding  galvanometer  de- 
flections being  noted,  the  scale  became  calibrated.  By  adjusting 
the  resistance  R  in  the  galvanometer  circuit,  the  value  of  the 
scale  could  be  varied  as  desired. 


FIG.  1310b. 

The  calibration  thus  briefly  outlined  was  worked  on  for  about 
three  weeks,  many  refinements  not  mentioned  here  were  used, 
hundreds  of  readings  were  recorded,  and  many  checks  made  upon 
the  observations  taken.  It  was  demonstrated,  as  a  net  result, 
that  this  apparatus  and  method  gave  continuous  temperature 
difference  records  that  were  not  in  error  over  0.01°  C.,  the  mean 
temperature  measured  being  about  —  37°  C.,  and  the  average 
difference  about  1.5°  C.  In  Fig.  1310b  is  given  on  a  much  re- 
duced scale  one  of  the  record  curves  obtained  in  a  run  of  12 
hours. 

1311.  The  Measurement  of  Extremely  High  Temperatures.  — 
Many  scientific  investigations  and  industrial  operations  now 
require  that  temperatures  shall  be  measured  at  which  all  materials 
deteriorate  or  become  fused  or  altered.  The  electrical  methods 
directly  applied  must  fail  to  be  of  service  here,  and  one  must  resort 
to  radiation  pyrometry.  The  various  methods  proposed  for 
measuring  high  temperatures  by  means  of  the  radiation  given  off 
from  a  hot  body  have  recently  received  much  study,  and  very 


ART.  1311]  MEASUREMENT   OF  TEMPERATURE  323 

successful  developments  have  followed  along  this  line.  The  sub- 
ject, however,  as  well  as  that  of  thermoelectric  pyrometry,  does 
not  fall  within  the  scope  of  this  work,  and  reference  must  be  made 
to  the  above-mentioned  treatise  of  Burgess  and  Le  Chatelier,  and 
to  an  excellent  summary  of  this  and  other  subjects  relating  to 
high-temperature  measurements  by  Dr.  C.  W.  Waidner.* 

*  "Methods  of  Pyrometry."     Printed  in  the  Proceedings  of  the  Engineers' 
Society  of  Western  Pennsylvania,  September,  1904. 


CHAPTER  XIV. 

INSTRUMENTS  USED  FOR  MEASURING  RESISTANCE. 
SOME  GENERAL  PRINCIPLES  CONSIDERED. 

1400.  Proposed  Treatment  of  Subject.  —  It  would  be  beyond 
the  scope  of  this  work  to  give  a  detailed  consideration  of  the 
requirements,  the  types,  the  design  and  the  construction  of  the 
instruments  used  in  the  measurement  of  resistance.     It  will  be 
profitable,  however,  to  consider  broadly  some  general  principles 
which  pertain  to  this  class  of  apparatus. 

The  apparatus  which  is  involved  chiefly  in  the  measurement  of 
resistance  consists  of: 

Resistance  standards  (medium,  low,  and  high),  resistance  boxes 
and  Wheatstone  bridges,  and  deflection  instruments  of  the  mirror 
and  pointer  types,  used  both  as  null  and  deflection  indicators. 
In  addition  to  the  more  essential  instruments  there  are  generally 
required  rheostats,  switches,  keys,  batteries,  etc.,  but  these  need 
no  further  mention  here. 

1401.  Conformity  in  the  Parts  of  an  Outfit.  —  In  selecting  a 
resistance-measuring  outfit,  when  there  is  latitude  of  choice,  the 
apparatus  should  be  chosen  so  its  various  parts  are  in  conformity. 
Thus,  if  resistance  boxes  and  the  samples  to  be  measured  have  a 
large  watt-dissipating  capacity,  then  a  galvanometer  of  moderate 
sensibility,  quick  in  action  and  robust  in  construction,  will  conform 
better  to  the  rest  of  the  equipment  than  a  delicate,  highly  sensitive 
instrument  which  deflects  slowly  and  requires,  perhaps,  repeated 
adjusting.     In  the  above  case,  if  the  magnitude  of  the  measur- 
ing current  is  properly   chosen,  a  relatively   insensitive  galva- 
nometer of  the  pointer  type  will  enable  the  samples  to  be  measured 
with  all  the  precision  which  may  be  obtained  from  the  rest  of  the 
apparatus,  while  the  speed  and  convenience  will  be  greater  than 
if  a  highly  sensitive  moving  magnet  galvanometer  with  lamp  and 
scale   is   selected.     On   the   other   hand,    if  the   samples   to   be 
measured,  as,  for  example,  tungsten  lamp-filaments,  are  small,  have 
a  high  temperature  coefficient,  and  are  incapable  of  dissipating 

324 


ART.  1402]      SOME  GENERAL  PRINCIPLES  CONSIDERED         325 

much  energy  without  heating,  then  the  galvanometer  must  be 
highly  sensitive.  But  if  the  galvanometer  is  sensitive  and  the 
samples  will  carry  little  current,  it  is  a  waste  of  room  and  expense 
to  use  a  Wheatstone  bridge  of  large  size  with  massive  brass  blocks 
and  with  resistance  coils  of  large  heat-dissipating  capacity. 

An  exercise  of  judgment  is  very  desirable,  in  regard  to  this 
matter  of  securing  conformity  in  the  various  parts  of  an  outfit. 

1402.  Sensibility  and  Accuracy.  —  The  sensibility  of  a  resist- 
ance-measuring outfit  is  determined  by  two  factors,  the  constant 
of  the  detector  or  deflection  instrument  and  the  magnitude  of  the 
measuring  current.  Both  of  these  are  limited,  but  the  limitation 
of  the  first,  when  there  is  latitude  of  choice,  is  rarely  reached. 
The  limitation  put  upon  the  second  may  result  from  the  limitations 
in  the  watt-dissipating  capacity  of  the  resistance  spools  in  the 
bridge  (generally  to  be  taken  as  0.5  watt  per  spool) ,  or  from  the 
watt-dissipating  capacity,  without  appreciable  heating,  of  the 
samples  which  are  measured.  No  approximation  to  a  rule  can 
be  given  for  this  last.  If  the  samples  have  a  negligible  tempera- 
ture coefficient,  they  can  dissipate  much  more  heat  without  chang- 
ing in  resistance  than  they  can,  other  things  equal,  if  they  have  a 
high  temperature  coefficient.  For  this  reason  much  greater  sensi- 
bility can  generally  be  obtained  in  measuring  the  resistance  of  the 
alloys  than  in  measuring  the  resistance  of  the  pure  metals.  It 
is  a  mistake,  however,  to  suppose  that  the  possible  sensibility  is 
much  less  with  a  low  resistance  of  a  given  length  of  sample  than 
with  a  high  resistance  of  the  same  length.  It  is  easily  seen  that, 
if  the  sample  is  in  the  form  of  a  ribbon  of  a  given  length,  doubling 
the  width  halves  the  resistance,  but  the  heat-dissipating  capacity 
is  also  doubled  and  hence  the  current  may  be  doubled.  The  result 
is  that  the  fall  of  potential  between  two  points  on  the  ribbon  can 
be  kept  constant  whatever  its  width.  Hence,  the  sensibility 
with  which  the  narrow,  high  resistance,  and  the  wide,  low  resist- 
ance, ribbon  may  be  measured  remains  the  same.  In  the  case  of 
round  wire,  as  used  in  a  platinum-resistance  thermometer  for 
example,  the  advantage  is  slightly  in  favor  of  the  fine  wire,  as  the 
cross-section  increases  with  increase  in  diameter  more  rapidly  than 
the  surface  from  which  the  heat  must  be  dissipated. 

In  general,  bridges  which  are  to  be  used  with  samples,  such  as 
resistance  thermometers  and  lamp  filaments,  may  have  very  small 
resistance  spools,  while  bridges  for  general  work  and  where  alloys 


326  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1403 

are  to  be  measured  should  have  larger  spools.  In  this  latter  case 
a  pointer  galvanometer  of  0.5  megohm  sensibility  will  serve  for 
most  requirements. 

The  absolute  precision  of  a  resistance  measurement  must  pri- 
marily depend  upon  the  accuracy  of  the  standards.  The  spools  in 
a  Wheatstone  bridge  become  for  the  time  being  the  standards 
employed.  It  is  always  possible,  if  one  has  one  standard  resist- 
ance, the  precision  of  which  is  known,  to  check  up  or  calibrate 
a  Wheatstone  bridge  so  that  the  true  resistance  of  all  its  coils 
becomes  known.  The  systematic  procedure  for  doing  this  will 
generally  be  given  to  customers  by  the  makers  of  the  bridges 
—  or  for  a  relatively  small  fee  a  bridge  will  be  tested  and  certi- 
fied to  by  the  National  Bureau  of  Standards. 

No  resistance  measurement,  however,  can  be  considered  as  very 
precise  unless  careful  attention  is  paid  to  the  magnitude  of  the 
measuring  current  and  the  temperature  of  the  surroundings  in 
which  the  sample  is  placed  at  the  time  it  is  measured.  For  much 
work  the  watt  capacity  of  the  bridge  spools  will  exceed  that  of 
the  samples  and  the  magnitude  of  the  current  must  be  adjusted 
to  the  latter. 

In  general,  it  is  considered  very  good  work  if  a  Wheatstone  bridge 
will  measure  resistances  from  10  to  10,000  ohms  with  a  precision 
of  0.02  of  1  per  cent.  Most  Wheatstone-bridge  work  will  range 
about  0.05  of  1  per  cent. 

1403.  Resistance  Standards.  — In  considering  resistance  units 
attention  should  be  drawn  to  the  distinction  between  resist- 
ance units  used  for  standards  of  resistance  and  resistance  units 
which  are  intended  to  carry  considerable  current  when  used  with 
potentiometers  and  like  instruments.  These  latter  should  be 
spoken  of  as  current-resistance  standards.  Resistance  standards 
proper  do  not  need  to  have  much  watt-dissipating  capacity. 
Their  requirements  are  unchangeableness  with  time,  low  tem- 
perature coefficient,  and  small  thermo-electromotive  force  against 
copper.  They  should  be  susceptible  also  of  immersion  in  kerosene 
oil,  so  their  temperature  may  be  accurately  ascertained.  The 
development  of  resistance  standards  has  passed  thru  quite  an 
evolution,  and  the  latest  type  as  designed  by  Dr.  E.  B.  Rosa  and 
endorsed  by  the  National  Bureau  of  Standards,  will  now  be  briefly 
described. 

A  paper  entitled  "The  Variation  of  Resistance  with  Atmospheric 


ART.  1403]      SOME  GENERAL  PRINCIPLES  CONSIDERED          327 


Humidity"  was  read  before  the  American  Physical  Society  at 
Washington,  April  21,  1907,  and  later  published  in  the  Bulletin  of 
the  Bureau  of  Standards,  Vol.  4,  1907-8,  page  121.  In  this 
paper  by  E.  B.  Rosa  and  H.  D.  Babcock,  it  was  demonstrated 
"  that  the  shellac  "  (covering  the  wire  of  wire-wound  resistance 
spools)  "  absorbs  moisture  from  the  surrounding  atmosphere  and 
expands,  stretching  the  manganin  wire  and  thereby  increasing 
the  resistance."  This  variation  in  resistance  under  circumstances 
exceptionally  favorable  for  stability  may  amount,  in  different 
seasons  of  the  year,  to  as  much  as  2  parts  in  10,000,  and  under  less 
favorable  circumstances  to  7  or  more  parts  in  10,000. 

The  discovery  of  this  property  of  shellac  of  swelling  with  mois- 
ture and  straining  the  wire  has  led  to  the  present  type  of  con- 
struction of  resistance  standards  adopted  by  the  Bureau  and 
executed  by  prominent  instrument  makers.  The  construction 
referred  to  is  fully  described  by  Dr.  E.  B.  Rosa  in  a  paper  entitled 
"  A  New  Form  of  Standard  Re- 
sistance," and  published  in  the 
Bureau  of  Standards'  Bulletin, 
Vol.  5,  1908-9,  page  413.  The 
essential  feature  of  the  construc- 
tion consists  in  hermetically  sealing 
the  brass  cylinder,  upon  which  the 
shellacked  resistance  wire  is  wound, 
in  a  metal  cylinder  filled  with  kero- 
sene oil.  In  this  manner  all  access 
of  moisture  to  the  resistance  wind- 
ing is  effectually  prevented.  This 
construction  so  improves  the  con- 
stancy of  the  resistance  that  of  28 
coils  kept  under  close  observation 
there  was  "  only  one  coil  of  the 
above  twenty-eight  that  has  changed 
as  much  as  two  parts  in  100,000 
during  the  past  twelve  months"  * 

This  type  of  standard,  for  re- 
sistance  units  of  10  ohms  or  more 
without  potential  terminals,  is  shown  in  section  in  Fig.  1403a. 

"  The  new  form  of  resistance  standard  is  much  smaller  than  the 
*  Bulletin,  Vol.  5,  page  427. 


FIQ 


328 


MEASURING  ELECTRICAL  RESISTANCE       [ART.  1403 


Reichsanstalt  type,  so  long  and  so  favorably  known  throughout  the 
world  as  a  standard  of  resistance.  It  weighs  only  about  400  g 
filled  and  measures  only  7.5  cm  across  the  terminals  instead  of 
16  cm.  For  measurements  up  to  an  accuracy  of  .001  per  cent 
it  is  measured  as  it  stands,  its  current  capacity  being  ample  when 
using  reasonably  sensitive  galvanometers,  and  the  small  ther- 
mometer in  the  central  tube  giving  its  temperature  with  all  needed 
accuracy.  The  temperature  coefficient  is  generally  not  greater 
than  .002  per  cent  per  degree,  so  that  a  quarter  of  1  degree  un- 
certainty in  the  temperature  would  cause  an  error  less  than  that 
allowed.  ******* 


FIG.  1403b. 

"  The  apparatus  employed  in  comparing  the  coils  of  any  denomi- 
nation with  one  another  is  shown  in  Figs.'7  1403b  and  1403c.  "It 
is  a  Wheatstone  bridge  with  the  coils  arranged  in  the  circle,  having 
two  extra  coils  inserted,  on  which  shunts  may  be  applied  for 
balancing  the  bridge  instead  of  shunting  directly  the  ratio  coils. 
Or  these  two  openings  may  be  closed  by  heavy  links  and  the 
shunts  applied  directly  to  the  ratio  coils.  The  circular  frame  is 
so  hinged  that  it  may  be  opened  far  enough  to  admit  a  larger 
coil,  as,  for  example,  one  of  the  Reichsanstalt  form,  which  may  thus 


AET.  1403]      SOME  GENERAL  PRINCIPLES  CONSIDERED         329 

be  directly  and  conveniently  compared  with  one  of  the  new  Bureau 
of  Standards  form,  Fig."  1403c.  "The  apparatus  is  very  conven- 
ient, is  compact,  requires  a  relatively  small  oil  bath,  or  may  be  used 
without  an  oil  bath  except  in  comparisons  of  extreme  precision, 
.and  will  accommodate  any  kind  of  a  resistance  standard  that  is 
provided  with  terminals  for  dipping  into  mercury  cups." 


FIG.  1403c. 

When  a  resistance  standard  is  1  ohm  or  less  it  is  necessary  to 
provide  it  with  potential  terminals.  This  leads  to  a  different 
construction  than  that  adopted  for  the  higher  denominations. 
The  resistance  material  for  these  low-resistance  standards  is  in  the 
form  of  heavy  manganin  wire  or  sheet  which  is  too  massive  to  be 
strained  by  the  swelling  of  shellac.  In  Fig.  1403d  is  shown  a 
0.001 -ohm  standard  resistance  of  the  Reichsanstalt  form. 

This  form  of  standard  is  guaranteed  by  its  makers  to  an  accu- 
racy of  0.02  of  1  per  cent  but  may  be  made  more  accurate. 
"  When  immersed  in  oil  and  used  as  a  precision  standard  of  resist- 


330 


MEASURING  ELECTRICAL  RESISTANCE       [ART.  1403 


ance  it  has  a  current-carrying  capacity  of  32  amperes.  Used 
for  measuring  current  to  a  lesser  degree  of  accuracy  it  will  carry 
100  amperes  or  more." 


FIG.  1403d. 

Otto  Wolff  of  Berlin  supplies  a  line  of  standard  resistances 
which  have  the  following  denominations  and  current-carrying 
capacities  when  used  for  precision  measurements  and  as  current 
resistance-standards . 


Resistance, 
ohms 

Current  capacity  for 
precision  measurements, 

Current  capacity  as 
current-resistance 

amperes 

standards,  amperes 

100,000 

0.003 

0.01 

10,000 

0.01 

0.03 

1,000 

0.03 

0.1 

100 

0.1 

0.3 

10 

0.3 

1 

1 

1 

3 

0.1 

3 

10 

0  01 

10 

100 

0  001 

30 

300 

0.0001 

100 

1000 

ART.  1404]      SOME  GENERAL  PRINCIPLES  CONSIDERED         331 

These  standards  are  called  "  Small  Pattern  "  standards.  They 
are  all  arranged  to  be  dipped  into  mercury  cups  by  means  of  copper 
terminals  which  in  all  types  are  separated  the  standard  distance 
of  16  cms  from  center  to  center.  The  resistance  proper  is  mounted 
inside  a  brass  case  with  a  hard  rubber  top  which  carries  the  ter- 
minals. The  case  is  perforated  to  permit  free  circulation  of  oil 
when  the  standard  is  immersed  in  a  bath  of  petroleum.  The 
general  rule  may  be  applied  that,  when  used  in  oil  for  high  pre- 
cision work,  a  load  of  1  watt  is  allowable,  while  for  less  accurate 
work  a  load  of  10  watts  may  be  applied. 

1404.  Resistance  Boxes  and  Wheatstone  Bridges;  General 
Remarks.  —  In  determining  the  value  of  one  resistance  in  terms 
of  another  it  is  very  desirable  to  have  a  series  of  known  values  of 
resistances  which  can  be  varied  thru  a  wide  range  by  known 
amounts.  The  fundamental  purpose  of  all  resistance  boxes, 
whether  employing  plug  contacts  or  dials,  is  to  provide  means  for 
obtaining  the  largest  possible  number  of  values  from  the  fewest 
possible  number  of  known  resistance  units.  The  very  varied 
forms  of  construction  that  instrument  makers  have  given  to  re- 
sistance boxes,  have  had,  more  or  less,  the  above  object  in  view. 
Where  the  number  of  coils  or  units  is  made  greater  than  the  least 
number  required  for  varying  resistance  thru  a  given  range,  it  is 
done  to  serve  some  purpose  of  convenience  of  working,  or  to  in- 
crease the  facility  with  which  .resistance  values  may  be  changed 
and  mentally  added  up. 

The  advantages  of  being  able  to  obtain  a  specified  number  of 
resistance  values  with  a  minimum  number  of  resistance  units 
pertain  both  to  the  user  and  to  the  maker  of  the  set.  The  ad- 
vantages to  the  user  are  reduced  cost,  a  smaller  number  of  coils  to 
get  out  of  adjustment  and  to  measure  when  checking  up  a  set, 
economy  of  space,  a  smaller  number  of  contacts  and,  often,  in- 
creased simplicity  in  forming  combinations  of  resistance  values. 

The  advantages  to  the  manufacturer  are  in  the  nature  of  re- 
duced cost  of  construction  which  must,  of  course,  be  reflected  to 
the  purchaser. 

In  short  it  is  because  of  the  gain  in  the  above  particulars  that 
it  is  desirable  to  combine  a  small  number  of  units  of  resistance  to 
obtain  a  great  many  values.  If  it  were  otherwise  there  would  be 
no  occasion  for  the  construction  of  resistance  boxes  and  one  would 
employ  a  separate  coil  for  every  value  which  he  might  wish  to  use. 


332  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1405 

The  various  methods  that  have  been  devised  for  combining  re- 
sistance units  have  been  discussed  already  and  it  may  only  be 
remarked  here  that  the  so-called  decade  plan  is  considered  so 
very  superior  to  all  others  that  its  further  adoption  is  urged. 

1405.  Watt  Capacity  of  Resistance  Units.  —  The  current- 
carrying  capacity  of  a  resistance  unit  in  a  resistance  set  will  de- 
pend not  only  upon  the  ability  of  the  unit  to  dissipate  heat  but 
also  upon  the  ohmic  resistance  of  the  unit.  It  is  therefore  meaning- 
less to  speak,  as  is  often  done  carelessly,  of  the  current  capacity 
of  coils.  The  watt-dissipating  capacity  of  a  coil  or  unit  is,  on  the 
other  hand,  a  definite  matter,  and  in  a  well-constructed  resistance 
set  will  be  the  same  for  each  coil  in  the  set.  The  watts  which  a 
spool  will  dissipate  will  be  approximately  proportional  to  the 
square  of  the  voltage  at  the  terminals  of  the  coil  divided  by  the 
resistance  of  the  coil;  or  it  will  be  approximately  proportional  to 
the  square  of  the  current  thru  the  coil  multiplied  by  the  resistance 
of  the  coil.  No  definite  statements  can  be  made  of  the  number 
of  watts  which  each  coil  in  a  set  may  safely  carry.  Often  a  tem- 
perature which  will  not  injure  the  insulation  will  permanently 
impair  the  precision  of  the  coil.  The  manganin  wire  of  which 
resistance  coils  are  mostly  wound  is  generally  permanently 
diminished  in  resistance  —  probably  by  the  release  of  the  molecu- 
lar strains  in  the  wire  —  when  the  temperature  is  considerably 
elevated.  To  make  this  effect  as  small  as  possible  resistance  coils 
should  be  artificially  "  aged  "  before  their  final  adjustment  by 
being  given  a  prolonged  baking  (from  24  to  48  hours)  at  a  tem- 
perature of  from  130°  to  140°  C.  It  is,  therefore,  unsafe  to  load  a 
precision  resistance  box  so  that  any  of  its  spools  attain  a  temper- 
ature above  that  at  which  they  were  aged.  A  general  rough  rule 
for  resistance  boxes,  the  spool  windings  of  which  are  on  brass,  is  to 
limit  the  watts  per  spool  to  3  watts.  Thus  one  should  limit  the 

v2 
current  thru  the  spools  so  that  — is  not  greater  than  3,  v  being  the 

volts  at  the  spool  terminals  and  r  the  resistance  in  ohms  of  its 
winding.  In  using  a  Wheatstone  bridge,  in  which  the  resistance 
of  the  ratio  arms  at  any  moment  may  be  one  time  1  ohm  and 
1000  ohms,  and  then  again  1000  ohms  and  1000  ohms,  it  is  well 
to  keep  from  10  to  100  ohms  in  the  battery  circuit  ali  the  time. 
An  E.M.F.  may  then  be  employed  which,  acting  thru  this  resist- 
ance, will  not  overheat  the  low-resistance  coils. 


ART.  1406]      SOME  GENERAL  PRINCIPLES  CONSIDERED 


333 


In  the  best  boxes  the  windings  are  not  only  upon  brass  spools 
but  these  are  in  direct  metallic  connection  with  the  brass  blocks 
upon  the  top  of  the  box.  By  this  arrangement  much  of  the  heat 
developed  is  conducted  to  the  brass  blocks  and  there  dissipated. 
Some  boxes  are  provided  with  outside  cases  of  perforated  metal 
so  the  entire  set  may  be  immersed  in  kerosene  oil.  A  set  used  in 
this  way  will  have  the  watt-dissipating  capacity  of  its  coils  in- 
creased several  times. 

It  is  rare,  however,  in  using  a  resistance  set  for  the  measurement 
of  direct-current  resistance  that  one  has  any  occasion  to  reach, 
even  approximately,  the  watt  capacity  of  the  set.  Plug  and  dial 
rheostat  boxes  and  sometimes  the  rheostats  of  Wheatstone-bridge 
sets  are  employed  as  auxiliary  apparatus  and  are  required  to 
carry  as  great  a  load  as  is  allowable.  For  this  reason  chiefly  it  is 
desirable,  in  selecting  resistance  sets,  to  require  that  their  heat- 
dissipating  capacity  shall  be  as  large  as  practicable  without  undue 
increase  in  the  size  of  the  sets. 

1406.  Construction  of  Resistance  Spools.  —  The  type  of  re- 
sistance spool  shown  in  the  illustration,  Fig.  1406,  combines  those 
features  which  experience  has  shown  to  be  desirable.  The  wind- 
ing is  upon  a  brass  tube.  The  wire  is 
chosen  of  such  a  size  (whenever  this  is 
possible)  that  the  required  resistance 
is  obtained  by  winding  it  in  a  single 
layer  which  extends  the  entire  length  of 
the  spool.  The  brass  spool  is  mounted, 
with  good  metallic  connections,  upon 
the  brass  shaft,  the  upper  end  of  which 
serves  as  a  stud  for  the  brushes  of  a 
dial  switch  to  rest  upon.  The  winding 
itself  is  of  manganin  wire  and  is  wound 
bifilar.  This  wire  is  chosen  because  of 
its  high  specific  resistance  (about  26 
times  that  of  copper),  small  tempera- 
ture coefficient  (about  0.00002  per 
degree  C.)  and  small  thermal  E.M.F. 
against  brass  or  copper.  There  are  other 
materials  which  would  probably  meet  the  above  conditions  quite 
as  well  but  they  are  of  recent  development  and  have  not  as  yet  had 
the  thoro  trial  and  endorsement  which  has  been  given  to  manganin. 


Brazed 
Copper 
Soldered 

FIG.  1406. 


334  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1407 

If  the  spools  are  100  ohms  or  less  in  resistance  the  terminals 
of  the  resistance-wire  proper  are  not  soft  soldered  directly  to  the 
brass.  A  short  length  of  copper  wire  is  silver  soldered  to  the 
manganin  and  this  terminal  is  in  turn  soft  soldered  to  the  brass. 
In  making  the  soft-soldered  connection  of  the  copper  terminals,  a 
small  variation  in  the  length  of  the  copper  terminal  affects  very 
little  the  total  resistance  of  the  spool.  Thus,  adjusted  spools  may- 
be soldered  in  place  in  resistance  sets  without  the  necessity  of 
making  a  further  resistance  adjustment  after  the  spool  is  in  place. 

The  prevailing  practice  is  to  cover  the  outside  of  the  spools  with 
a  thick  covering  of  shellac  which  is  baked  hard  and  is  glossy.  As 
shellac  has  been  shown  to  absorb  moisture  from  the  air  and  to 
strain  the  wire  by  swelling,  a  non-hydroscopic  material  should 
be  sought.  If  the  proper  material  were  found  the  permanent 
precision  of  resistance  sets  would  be  considerably  improved. 

Spools  constructed  and  mounted  as  above  may  be  made  to 
dissipate  safely  between  three  and  four  watts  each.  In  boxes  of 
cheaper  grade  the  manganin  is  wound  upon  wooden  spools.  These 
are  inferior  in  heat-dissipating  capacity  and  other  respects.  When 
of  the  same  size  as  the  brass  spools  their  watt-dissipating  capacity 
is  between  0.25  and  0.5  watt  or  about  0.1  that  of  a  brass  spool 
mounted  as  above. 

1407.  The  Precision  of  Coils  in  Resistance  Sets.  —  The  pre- 
cision of  coils  in  resistance  sets  is  not  usually  required  to  be  so 
high  as  that  of  individual  resistance  standards.  Even  if  the 
adjustments  were  made  originally  as  high  as  that  of  the  best 
individual  standards  they  would  only  hold  for  a  particular  tem- 
perature. Furthermore,  as  different  sizes  of  wire  must  be  used 
in  the  same  resistance-set,  it  is  impracticable  to  select  all  the  sizes 
of  exactly  the  same  temperature  coefficient;  and  thus  exact  tem- 
perature corrections  for  all  the  coils  in  a  set,  for  other  tempera- 
tures than  that  at  which  they  were  adjusted,  are  hardly  possible. 
The  coils,  moreover,  not  being  hermetically  sealed  in  oil  will  not 
maintain  the  high  precision  which  might  be  given  to  them  and 
which  pertains  to  resistance  standards.  The  usually  unknown 
resistance  of  leads,  the  contact  resistances  under  binding  posts, 
and  the  various  plug  or  brush  contact  resistances  which  enter  in 
a  resistance  set  is  another  reason  why  the  highest  possible  pre- 
cision is  not  demanded.  The  best  makers  guarantee  conserva- 
tively the  precision  which  pertains  to  the  resistance  sets  which 


ART.  1408]      SOME   GENERAL  PRINCIPLES  CONSIDERED         335 

they  list.  A  Wheatstone  bridge  of  the  highest  grade,  intended  for 
precision  work,  will  have  the  following  guarantee:  The  rheostat 
coils  will  be  guaranteed  to  an  accuracy  of  0.02  of  1  per  cent,  with 
the  exception  of  the  0.1-ohm  coils  which  will  be  guaranteed  to  0.1  of 
1  per  cent  and  the  1-ohm  coils  to  0.04  of  1  per  cent.  The  coils  hi  the 
ratio  arms  of  such  a  set  will  be  guaranteed  to  be  adjusted  to  an 
accuracy  of  0.02  of  1  per  cent  and  to  be  like  each  other  to  0.01  of  1 
per  cent.  A  Wheatstone  bridge  of  more  moderate  precision  and  of 
about  one- third  the  cost  would  be  given  an  accuracy  of  adjustment 
of  the  coils  in  the  rheostat  of  0.05  of  1  per  cent  and  in  the  ratio 
arms  of  0.02  of  1  per  cent.  The  very  cheapest  boxes  and  Wheat- 
stone  bridges  should  not  be  less  accurate  than  0.2  of  1  per  cent  in 
their  rheostat  and  0.1  of  1  per  cent  in  their  ratio  arms. 

1408.   Some  Features  of  Outside  Construction.  —  The  classical 
method,  for  many  years  the  only  one  employed,  of  making  the 


FIG.  1408a. 

desired  combinations  of  resistance  coils,'  is  to  use  brass  blocks 
mounted  on  hard  rubber  plates,  which  are  connected  by  plugs 
inserted  between  them.  In  recent  years  other  methods  have  come 
more  and  more  into  favor,  especially  where  it  is  required  to  obtain 
the  desired  values  rapidly.  Among  such  of  special  value  is  the 
method  of  dial  contacts. 

The  chief  advantage  in  the  use  of  plug  contacts  consists  in  the 
great  number  of  combinations  which  can  be  effected  with  com- 
paratively few  blocks  and  plugs.  The  advantage  has  long  been 
claimed  for  plug  contacts  that  this  form  of  contact  has  the  least 
resistance.  This  advantage,  however,  cannot  be  allowed  for  plug 


336 


MEASURING  ELECTRICAL  RESISTANCE       [ART.  1408 


contacts  if  certain  other  forms  of  contacts  are  correctly  designed 
and  constructed. 

In  resistance  sets  of  the  plug-type  it  is  desirable  to  use  a  correct 
type  of  plug.  This  should  have  a  proper  taper  to  fit  the  holes 
between  the  blocks.  The  hard  rubber  head  should  be  easy  to 
grasp  without  hurting  the  fingers  and  should  not  work  loose. 
The  type  of  plug  designed  by  the  author,  and  shown  in  Fig.  1408a 
meets  these  conditions  and  has  been  much  used  on  high-grade  sets. 

Brass  blocks  when  used  should  be  of  sufficiently  heavy  con- 
struction to  permit  undercutting  so  the  base  of  the  blocks  will  be 
separated  to  give  sufficient  surface  insulation  over  the  hard  rubber 
top  of  the  resistance  set.  This  is  important  because  hard  rubber, 
which  is  exposed  to  light  for  a  long  time,  deteriorates  in  its  sur- 
face insulation. 


FIG.  1408b. 


In  Fig.  1408b  is  shown  a  high-grade  Wheatstone-resistance  set  of 
the  plug  decade  type  which  is  intended  to  embody  all  of  the  best 
and  latest  features  of  Wheatstone-bridge  construction.  This  set 
is  of  the  four-coils-to-the-decade  type  as  described  in  par.  503. 
An  extra  coil  is  added  to  the  decade  of  lowest  denomination.  This 
extra  coil  is  added  to  enable  the  set  to  be  checked  up.  It  is  pos- 
sible to  compare  every  coil  in  a  set  with  every  other  coil  in  the  set, 
and  to  completely  intercheck  the  different  decades. 

A  particularly  fine  type  of  a  five-dial  Wheatstone  bridge,  con- 
structed for  the  author,  embodies  the  following  special  features 


ART.  1408]      SOME  GENERAL  PRINCIPLES  CONSIDERED         337 

of  construction:  The  case  of  the  bridge  is  made  of  glass  which 
gives  a  full  view  of  the  coils  and  inside  wiring.  The  rheostat 
consists  of  five  dials,  the  one  of  lowest  denomination  reading  in 
0.1  of  an  ohm.  The  ratio  arms  can  be  plugged  to  give  ratios  1  to 
10,000  or  10,000  to  1  with  many  other  ratios  lying  between  these 
extremes.  The  four-coils-to-the-decade  construction  is  used  in 
the  rheostat  (see  §  503)  and  Schone's  arrangement  is  used  in 
the  ratio  arms  (see  §  509).  The  accuracy  of  adjustment  of  the 
coils  in  this  bridge  is  better  than  0.02  of  1  per  cent  in  the  rheostat 
and  still  better  in  the  ratio  arms. 


FIG.  1408c. 

The  form  of  dial  construction  which  perhaps  has  met  with  most 
favor  is  the  one  used  by  Otto  Wolff  of  Berlin  in  all  his  resistance 
sets  and  potentiometers.  A  top  view  of  an  Otto  Wolff  five-dial 
Wheatstone  bridge  is  shown  in  Fig.  1408c. 

Dial  switches  are  only  employed  for  the  rheostat,  the  values  in 
the  ratio  arms  being  obtained  with  plugs  inserted  in  blocks. 


CHAPTER  XV. 
DEFLECTION  INSTRUMENTS  AND  GALVANOMETERS. 

1500.  Distinction  Between  Indicators  and  Deflection  Instru- 
ments. —  We  shall  discuss  these  chiefly  from  the  standpoint  of 
their  employment  in  resistance  measurements. 

As  resistance  measurements  may  be  broadly  divided  into  de- 
flection methods  —  in  which  the  value  sought  is  determined  in 
terms  of  the  deflections  obtained  with  a  deflection  instrument 
-  and  null  methods  —  in  which  a  current  detector  is  used 
merely  to  indicate  when  no  current  is  flowing  in  its  circuit  — 
the  instrumental  requirements  for  the  two  methods  are  quite 
dissimilar. 

For  deflection  methods  the  instruments  used  are  voltmeters 
and  ammeters,  pointer  galvanometers,  and  mirror  galvanometers 
with  telescope  and  scale  or  lamp  and  scale. 

When  deflection  instruments,  voltmeters  and  pointer  galva- 
nometers are  used  for  measuring  resistance  it  is  required  that  the 
pointer  should  accurately  return  to  zero  after  being  deflected  and 
that  the  scale  be  as  long  as  practicable.  This  should  be  either 
carefully  calibrated  for  correctly  measuring  equal  increments  of 
current,  or  if  the  scale  divisions  are  equally  spaced  the  deflections 
should  increase  proportionally  with  the  current.  If  the  above 
requirements  are  met,  then  the  possible  precision  obtainable  will 
increase  with  the  number  of  smallest  divisions  marked  upon  the 
scale. 

The  damping  of  the  system  of  such  a  deflection  instrument 
should  be  such  that  the  deflection  will  reach  quickly  its  maximum 
value  without  any  oscillation.  It  is  much  easier  to  take  close 
readings  —  when  the  measuring  current  is  not  very  steady  —  if 
the  instruments  are  aperiodic  or  critically  damped. 

It  is  not  generally  required  that  deflection  instruments,  when 
used  for  measuring  resistance,  shall  have  a  very  high  sensibility,  ex- 
cept when  used  for  insulation  testing  by  direct  deflection  methods. 
A  good  voltmeter  of  the  Weston  type  will  have  about  100  ohms 

338 


ART.  1500]  DEFLECTION  INSTRUMENTS  339 

to  the  volt  and  consequently,  for  full  scale  deflection,  will  take 

about  TT-7T™  =  0.01  ampere.     A  current  of  this  magnitude  will 
JLOjUUU 

cause  very  little  heating  in  any  resistance  which  is  measured  — 
tho  even  this  small  current  will  alter  the  resistance  of  a  tungsten 
lamp-filament  which,  consequently,  cannot  be  measured  accurately 
with  a  voltmeter.  One  need  for  sensibility  in  such  an  instrument 
is  to  be  able  to  use  so  small  a  measuring  current  that  resistances 
will  not  be  appreciably  heated. 

Portability  is  a  feature  which  is  generally  sought  and  is,  of 
course,  obtained  with  all  instruments  of  the  voltmeter  or  am- 
meter type. 

For  measuring  insulation  resistance  by  direct  deflection  specially 
designed  and  constructed  galvanometers  which  have  high  sensi- 
bility and  proportional  deflections  are  required.  These  high 
sensibility  insulation-testing  galvanometers  were  formerly  of  the 
moving-magnet  type,  the  typical  instrument  being  the  mirror 
galvanometer  of  Lord  Kelvin.  The  improvements  which  have 
been  made  recently  in  moving-coil  or  D'Arsonval  galvanometers 
have  been  such  that  these  are  now  much  more  generally  used. 
Their  sensibility  is  sufficient  for  most  classes  of  work  and  they 
are  much  more  convenient  to  use.  They  have  a  stable  zero,  their 
deflections  are  nearly  proportional  over  a  scale  a  meter  long  set 
at  two  meters  from  the  mirror,  and  they  may  be  made  portable 
for  outdoor  work.  The  best  of  these  instruments  represent  the 
highest  art  in  galvanometer  design.  We  shall  discuss  later  their 
more  prominent  characteristics. 

!  The  instrument  most  universally  employed  in  resistance  measure- 
ments, made  by  any  null  method,  is  a  galvanometer.  Galvanom- 
eters are  so  varied  in  form  and  character  that  we  can  only  consider 
a  few  of  the  most  prominent  facts  regarding  them. 

The  system  of  a  pointer  instrument  or  mirror  instrument,  whether 
of  the  moving-magnet,  or  the  moving-coil  type,  is  essentially  an 
oscillating  system.  But  in  resistance  measurements — where  the 
null  method  is  used  —  the  possible  amplitude  of  the  oscillation 
need  not  be  large  and  proportionality  of  the  deflections  is  not 
important.  In  fact,  if  a  galvanometer  is  to  be  used  expressly  for 
measuring  resistance  by  a  null  method,  the  movement  of  its 
system  may  be  limited  by  stops  as  there  is  no  necessity  that  its 
pointer  or  mirror  should  move  thru  more  than  a  very  small  angle 


340  MEASURING  ELECTRICAL  RESISTANCE      [ART.  1501 

to  indicate  when  current  is  passing.  This  simplifies  the  design 
of  zero  indicators,  for  the  rather  difficult  requirement  of  giving 
the  instrument  proportional  deflections  over  a  long  scale  does  not 
need  consideration. 

Zero-current  indicators  or  galvanometers  of  a  portable  type 
which  have  a  pointer  that  moves  in  either  direction  from  the 
center  over  a  scale  of  10  or  20  divisions  can  be  obtained  that  are 
portable  and  sufficiently  sensitive  for  the  great  majority  of  meas- 
urements of  resistance  between  1  and  100,000  ohms. 

Very  many  resistance  measurements  are  made  with  a  slide  wire 
or  Wheatstone  bridge  in  factories  and  colleges  where  a  pointer 
galvanometer  would  be  found  amply  sensitive  and  far  more 
convenient  to  use  than  the  suspended  coil,  mirror  type  of  gal- 
vanometer so  generally  employed.  A  good  pointer  galvanometer 
of  300  ohms  resistance  should  deflect  over  its  scale  1  mm.  with 
1  volt  and  with  0.5  to  1  megohm  in  series.  It  should  be  just  short 
of  critically  damped,  and  come  to  rest,  after  the  key  is  closed,  in 
from  two  to  three  seconds.  This  type  of  instrument  —  which  can 
be  obtained  at  half  the  cost  of  the  better  types  of  suspended  coil 
instruments  —  is  strongly  advocated  for  the  use  of  elementary 
laboratory  students.  It  will  be  found  suitable  for  all  general 
Wheatstone-bridge  work  not  requiring  accuracies  of  over  0.05  to 
0.1  of  1  per  cent.  Such  instruments  are  made  by  Weston,  Paul, 
of  London,  and  others. 

The  author  also  has  given  much  attention  to  the  development 
of  this  type  of  galvanometer  and  the  following  is  a  brief  descrip- 
tion* of  the  new  type  which  he  designed  and  which  is  now  marketed 
by  the  Leeds  and  Northrup  Co. 

1501.  Pointer  Type,  Flat-coil  Galvanometers.  —  The  gal- 
vanometer is  called  a  flat-coil  galvanometer  and  is  a  somewhat 
radical  modification  of  the  moving  coil  instrument.  The  modi- 
fication was  made  by  the  author  to  make  the  moving  coil  instru- 
ment into  a  sensitive,  substantial  and  portable  galvanometer  in 
which  suspensions  might  replace  the  customary  jewels.  A 
suspended  system  may  be  arranged  to  withstand  rougher  handling 
than  a  system  mounted  in  jewels,  and  there  is  the  further 

*  For  a  more  detailed  description  of  this  and  other  galvanometers  consult 
a  paper  by  the  author  in  the  Journal  of  the  Franklin  Inst.,  October,  1910, 
entitled,  "The  Comparison  of  Galvanometers  and  a  New  Type  of  Flat-coil 
Galvanometer,"  from  which  paper  much  of  what  is  given  here  is  taken. 


ART.  1501] 


DEFLECTION  INSTRUMENTS 


341 


marked  advantage  in  a  suspended  instrument  that  there  is  noth- 
ing which  corresponds  to  an  initial  pivot  friction.  If  very  minute 
deflections  of  a  pivot  instrument  are  optically  magnified,  they 
will  be  found  to  be  erratic  and  in  no  wise  proportional  to  the 
forces  acting.  A  suspended  instrument,  on  the  other  hand,  will 
deflect  somewhat  with  the  feeblest  force,  and  if  its  minute 
deflections  are  highly  magnified  by  optical  means,  they  will  be 
found  to  be  proportional  to  the  deflecting  force.  This  circum- 
stance especially  adapts  a  suspended  instrument  to  null  measure- 
ments. Furthermore,  the  disk-like  shape  of  the  moving  system 
lends  itself  in  a  portable  instrument,  with  a  suspended  system, 
to  a  construction  which  requires  a  comparatively  small  height. 

Section  on  A-B 


f 

3 
-6 

11 

3  i 

<- 

C  1 

>. 

l* 

Guard 

Soft  Iron        -•» 
Pole  Pieces 


System    Guard-* 


Spring  Terminal  for 
Jjower  Suspension 


FIG.  1501a. 


Fig.  1501a  shows  the  construction  of  one  type  of  this  instru- 
ment, used  as  a  pointer  instrument  which  is  portable.  The  cut  is 
made  complete  and  explicit  to  save  lengthy  description.  The 
same  instrument  as  it  appears  when  mounted  in  its  case  is  shown 
in  Fig.  1501b. 

It  will  be  observed  that  when  current  energizes  the  four  disk- 
like  coils  they  all  contribute  to  produce  rotation  in  the  same 
direction.  A  reversal  of  the  direction  of  the  current  reverses  the 
direction  of  rotation,  the  deflections,  with  equal  current,  being  the 
same  to  either  side  of  the  zero. 


342 


MEASURING  ELECTRICAL  RESISTANCE       [ART.  1501 


At  first  sight  this  form  of  system  might  seem  bad  for  giving  a 
high  figure  of  merit  because  so  large  a  portion  of  the  moving  parts 
are  far  removed  from  the  axis.  While 
this  is  true,  there  are  compensating  ad- 
vantages which  practice  shows  make  the 
galvanometer  very  creditable  in  respect 
to  figure  of  merit,  while  many  features 
are  secured  which  have  practical  worth. 
In  the  first  place,  as  the  galvanometer 
has  no  iron  core,  there  are  only  two  air 
gaps,  one  above  and  one  below  the 
coil,  instead  of  four,  as  required  by  the 
D' Arson  val  galvanometer  with  an  iron 
core;  in  the  second  place  all  of  the  mag- 
netic field  is  active  at  all  times  in  pro- 
ducing a  turning  moment,  regardless  of 
the  angular  position  of  the  system.  In 
the  D'Arsonval  type,  a  magnetic  field 
must  be  provided  for  the  system  to  deflect  into.  This  is  shown 
by  Fig.  1501c. 


FIG.  1501b. 


FIG.  1501c. 


Thus  if  the  coil  section  C\  occupies  one  fifth  of  the  air  gap,  or 
space  in  which  the  coil  turns,  .then  four  fifths  of  the  magnetic 
lines  are  at  all  times  idle.  Magnets  of  sufficient  strength  and 
capacity  must  be  supplied  to  furnish  five  times  as  many  magnetic 
lines  as  are  at  any  one  moment  used.  The  flat-coil  galvanometer 
has  a  large  advantage  in  this  respect. 

Moreover,  the  aluminum  form  in  which  the  four  coils  are  held 
serves  admirably  as  a  magnetic  damper.  Further,  if  the  instru- 
ment is  to  serve  as  a  ballistic  galvanometer,  its  moment  of  inertia 
can  be  made  very  great  out  of  material  which  is  active  in  produc- 
ing a  turning  moment  and  for  such  service  the  construction  is 
ideal. 


ART.  1501]  •     DEFLECTION  INSTRUMENTS  343 

The  best  proportioning  of  the  system  and  coils  to  give  specific 
results  has  received  at  the  hands  of  the  author  much  thought  and 
calculation. 

The  accessory  features  connected  with  this  galvanometer,  as 
magnet,  suspensions,  etc.,  deserve  a  few  remarks. 

The  galvanometer  is  mounted  in  a  small  wooden  box  and  is 
very  portable.  For  ease  in  shipping  and  in  manufacture  cast- 
iron  magnets  are  used.  These  are  very  suitable,  and  if  properly 
proportioned  they  give  a  field  which  is  as  strong  as  can  be  used 
with  advantage  in  a  galvanometer  in  which  the  sensibility  is  made 
large  by  the  use  of  a  weak  suspension.  There  are  always  magnetic 
impurities  in  the  coil  system.  The  magnetic  field  in  which  the 
coil  swings  acts  on  these  impurities,  probably,  in  rough  proportion 
to  the  square  of  the  field  strength.  The  result  is,  that  the  system 
takes  on  a  polarity  and  tends,  irrespective  of  the  suspension,  to 
set  in  a  particular  angular  position.  To  show  this  strikingly, 
place  the  poles  of  a  fairly  strong  permanent  U-magnet  near  the 
poles  of  an  ordinary  wall  form  D'Arsonval  galvanometer  of  about 
400  megohms  sensibility.  It  will  generally  show  a  permanent 
deflection  of  from  10  to  20  scale  divisions  as  long  as  the  U-magnet 
is  held  on. 

The  natural  zero  of  the  galvanometer  is  the  position  taken  by 
the  coil,  which  position  is  the  resultant  of  the  torsional  force  of 
the  suspension,  and  the  magnetic  action  of  the  field  upon  the 
magnetic  impurities  in  the  coil.  When  the  field  is  weakened  or 
changed  in  direction,  the  resultant  controlling  force  changes,  and 
hence  a  deflection  results.  In  attempting  an  accurate  measure- 
ment, by  the  Kelvin  double  bridge,  of  the  resistance  of  a  3000- 
ampere  standard  low  resistance,  the  author  recently  used  a 
D'Arsonval  galvanometer  in  close  proximity  to  one  of  the  con- 
ductors carrying  1000  amperes.  .Whenever  the  circuit  thru  the 
low  resistance  was  closed  the  galvanometer  deflected  one  scale 
division  by  the  influence  of  the  field  external  to  the  conductor. 
This  deflection  would  have  produced  an  error  in  measuring  the 
resistance  of  0.1  of  1  per  cent,  if  it  had  not  been  discovered. 
The  galvanometer  was  a  very  good  one  and  the  circumstance  is 
related  to  show  how  precautions  against  exterior  fields  must  be 
taken  sometimes,  even  with  moving-coil  galvanometers.  It  also 
shows  that  little,  if  any,  improvement  can  be  obtained  in  weakly 
controlled  galvanometers  by  using  magnetic  fields  greater  than 


344  MEASURING  ELECTRICAL   RESISTANCE       [ART.  1501 

can  be  obtained  with  cast-iron  magnets.  It  may  be  mentioned 
here,  also,  that  the  chief  cause  of  zero  shift  in  sensitive  D'  Arson  val 
galvanometers  is  due,  not  to  the  "  set  "  in  the  suspension,  but  to 
a  "  magnetic  set  "  which  the  coil  takes  when  deflected  in  a  strong 
field. 

If  a  galvanometer  is  to  have  a  quick  period  and  a  strong  con- 
trol, and  be  magnetically  damped,  then  it  is  desirable  to  use  a 
magnetic  field  of  high  intensity,  such  as  can  be  secured  only  by 
using  soft  Swedish  iron  pole  pieces  and  permanent  steel  magnets 
of  the  best  quality. 

Galvanometers  which  carry  a  pointer,  or  a  mirror,  and  are 
intended  for  use  on  shipboard  or  for  easy  transportation  and 
setting  down  without  leveling,  must  be  suspended  between  upper 
and  lower  suspensions  which  are  taut.  Guards  which  limit  the 
movement  of  the  system  in  all  directions,  together  with  a  spiral 
spring  in  both  upper  and  lower  suspension,  will  effectually  prevent 
the  suspensions  breaking,  even  when  the  galvanometer  is  sub- 
jected to  severe  jars. 

It  is  desirable,  however,  to  make  the  tensile  strength  of  the 
suspensions  great  without  undue  increase  in  their  torsional  moment. 
The  tension  on  the  suspensions  may  then  be  made  considerable  and 
it  becomes  easy  to  accurately  balance  the  system.  This  accurate 
balancing  is  essential  to  a  steady  zero  and  to  make  the  system  free 
from  vibrations.  A  system,  held  between  two  taut  suspensions, 
which  is  slightly  unbalanced  about  its  axis  of  rotation  is  much 
more  influenced  by  tremors  than  one  which  is  carefully  balanced. 

The  tensile  strength  of  the  suspension  can  be  increased  greatly 
without  an  increase  in  its  torsional  moment  by  making  it  of  a 
number  of  round  wires  laid  together  in  a  cable,  instead  of  a  single 
wire.  Thus  : 

Let  n  =  the  number  of  wires  laid  together, 
d  =  the  diameter  of  a  single  wire. 


Then  the  torsion  of  one  wire  is  tocd^xS2,  where  S  is  the  cross- 
section  of  one  wire.     The  total  torsion  of  the  n  wires  is 


If  these  n  wires  were  to  be  combined  into  one,  the  tensile  strength 
would  remain  the  same,  but  the  torsion  would  now  be 

T8  oc  (nS)2. 


ART.  1501] 


DEFLECTION   INSTRUMENTS 


345 


Hence, 


T_ 
Ta 


nS* 


(1) 


This  conclusion,  that  with  equal  tensile  strength  the  moment 
of  torsion  decreases  with  the  number  of  strands  into  which  a 
given  cross-section  of  wire  is  subdivided,  has  great  practical 
value.  It  is  much  employed  in  galvanometer  constructions  where 
the  galvanometers  are  to  be  made  portable.  Experience  shows 
that  there  is  no  slipping  of  one  strand  over  the  other  in  a  manner 
to  develop  friction  and  uncertain  return  to  zero. 

The  flat-coil  type  galvanometer  here  described  is  admirably 
suited  to  a  differential  instrument,  especially  for  use  in  tem- 
perature measurements  with  resistance  thermometers.  Its  appli- 
cations to  this  service  have  been  pointed  out  by  the  author  in  a 
former  publication.* 

When  used  as  a  deflection  galvanometer,  its  deflections  are  not 
as  proportional  to  the  current  flowing  as  might  be  desired.  Fig. 
1501  d  is  a  curve  which  gives  the  average  performance  of  the 
instrument  in  this  respect. 


+10 


50 


100 


150  200  250 

Divisions  Deflection. 

FIG.  1501d. 


350 


400 


This  type  of  galvanometer  may  be  given  any  resistance  from 
20  ohms  to  300  ohms.  One  made  with  a  steel  magnet  and  of 
300  ohms  resistance  will  have  a  sensibility  of  about  one  millimeter 
deflection  upon  its  own  scale  with  1  volt  and  1  megohm  in  series, 
when  the  period  for  one  complete  oscillation  of  its  system  is  1.5 

*  "  Cooling  Curves  and  a  New  Type  of  Apparatus  for  their  Automatic 
Registration,"  Proc.  of  the  Amer.  Electrochem.  Soc.,  May  7,  1909. 


346  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1502 

seconds.     The  system  is  magnetically  damped  by  its  aluminum 
coil  frame  to  be  slightly  less  than  just  aperiodic. 

1502.  Sensitive  Galvanometers  for  Refined  Measurements  of 
Resistance  and  Insulation  Testing.  —  For  precise  measurements 
of  low  resistance,  using  a  Kelvin  double  bridge,  and  for  measuring 
the  insulation  resistance  by  direct  deflection  the  pointer  type  of 
galvanometer  has  insufficient  sensibility  and  some  type  of  mirror 
galvanometer  with  a  suspended  system  is  usually  required. 

To  aid  in  the  selection,  understanding  and  use  of  galva- 
nometers to  meet  the  above  requirements  we  shall  now  give  a 
brief  outline  of  the  fundamental  principles  upon  which  these 
instruments  operate.  Some  of  the  principles  will  also  be  dis- 
cussed which  will  assist  in  making  a  rational  comparison  of  the 
merits  and  usefulness  of  the  very  many  different  types  and 
forms  of  mirror  galvanometers  which  the  instrument  maker 
supplies. 

1503.  The  Equation  of  Motion  of  a  Galvanometer  System. 
—  The  systems  of  all  galvanometers  may  be  considered,  and  are 
to  be  mathematically  treated,  as  oscillating  systems.     The  move- 
ments of  such  systems  obey  the  same  laws  and  are  expressed  by 
the  same  equation  of  motion  as  the  oscillations  of  a  pendulum 
bob  which  is  damped  in  its  movements  by  swinging  in  a  viscous 
fluid.     When,  also,  the  electricity  in  an  electric  circuit,  that  con- 
tains a  condenser,  a  self-inductance,  and  an  ohmic  resistance,  is 
allowed  to  oscillate  freely  its  motion  can  be   expressed  by   an 
equation  of  the  same  form  as  expresses  the  oscillations  of  a  volt- 
meter system.     Thus  it  is  that  the  solutions  which  are  given  in 
many  treatises  and  textbooks  for  the  oscillations  of  electricity 
may  be  applied  —  by  a  simple  change  of  constants  —  to  the 
movements  or  oscillations  of  the  moving  systems  of  all  types  of 
deflection  instruments.     Those,  therefore,  who  desire  to  study 
the  laws  which  govern  the  motions  of  deflection  instruments  will 
do  well  to  keep  the  above  analogy  in  mind  and  study  some  one 
of  the  many  treatments  which  are  very  fully  given  for  the  elec- 
trical case. 

The  equation  of  motion,  which  governs  all  freely  oscillating 
systems,  when  put  in  a  form  that  applies  to  a  galvanometer  system 
is  expressed  as  follows: 

*+*+« -o.  a) 


ART.  1503]  DEFLECTION  INSTRUMENTS  347 

Here 

6  =  the  angle  thru  which  the  system  has  turned, 
t  =  the  time, 

/  =  the  moment  of  inertia  of  the  system, 
B  =  a  constant  called  the  coefficient  of  damping,  and 
*  K  =  a  constant  called  the  moment  of  torsion. 

The  analogous  equation  for  the  motion  of  electricity  in  a  cir- 
cuit is 

*f+«f+^H>,  eo 

where 

q  =  the  quantity  of  stored  electricity, 
t  =  the  time, 

L  =  the  coefficient  of  self-induction  of  the  circuit, 
R  =  a  constant  —  the  ohmic  resistance  of  the  circuit  —  and 

-^  =  a  constant  —  the  reciprocal  of  the  capacity  of  the  circuit. 
C 

The  forms  of  Eqs.  (1)  and  (2)  being  identical  their  solutions 
will  be  the  same  with  the  exception  of  the  arbitrary  constants 
involved. 

It  is  not  our  purpose  to  give  here  the  solutions  of  these  equa- 
tions, but  merely  to  point  out  the  useful  conclusions  to  be  drawn 
from  them.  Readers  who  are  interested  in  the  mathematical  side 
of  the  subject  are  referred  to  the  following  books  and  papers: 

Lamb's  "  Infinitesimal  Calculus,"  page  512  and  preceding  pages. 
"  Instruments  et  Methodes  de  Mesures  Electriques  Industrielles  par 
H.  Armagnat,  1902  edition."  Physical  Review,  March,  1903, 
page  158,  where  is  contained  an  excellent  article  by  O.  N.  Stewart 
on  "  The  Damped  Ballistic  Galvanometer."  Also,  "  Alternating 
Currents,"  by  Bedell  and  Crehore.  As  shown  above,  all  facts 
deduced  by  the  two  latter  authors  from  the  equations  of  motion 
concerning  the  electrical  case  apply,  with  modification  of  the 
constants  only,  to  the  mechanical  problem. 

Some  deductions  to  be  drawn  from  the  differential  equations 
and  their  solutions  may  be  considered  as  follows:  Suppose  we 
substitute  for  the  scale  a  photographic  film  which  is  drawn  along 
in  a  vertical  direction  at  a  uniform  speed  while  a  spot  of  light  is 
reflected  from  a  mirror  on  the  coil  upon  this  film,  it  being  assumed 


348  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1503 

that,  when  the  coil  deflects,  the  spot  of  light  moves  over  the  film 
in  a  horizontal  direction.  If  the  coil  has  been  deflected  in  any 
manner  thru  an  angle  6m  and  then  left  to  swing  freely,  the  spot  of 
light  will  trace  on  the  photographic  film  a  certain  curve.  The 
form  which  this  curve  will  have  will  be  entirely  dependent  upon 
the  values  given  to  the  constants  /,  B,  K,  in  Eq.  (1).  • 

In  Fig.  1503  are  given  some  characteristic  curves  which  might 
be  obtained  in  the  above  way,  and  which  show  the  effect  of  varying 

the  magnitude  of  the  constants. 

The  movement  of  the  system  is 
periodic  if  it  passes  the  zero  position, 
or  position  of  equilibrium.  This  case 
is  illustrated  by  curves  I,  II  and  III. 
If  there  is  no  damping,  that  is,  if  the 
constant  B  is  zero,  the  system  oscil- 
lates indefinitely  and  equal  distances 
each  side  of  the  zero.  This  is  illus- 
trated by  curve  I.  If  B  is  small,  but 
not  zero,  the  damping  is  only  par- 
tial, as  curve  II  best  illustrates,  the 
FIG.  1503.  magnitude  of  the  oscillations  gradu- 

ally decreasing. 

For  a  certain  critical  value  of  B  the  system  returns  with  the 
maximum  speed  possible  to  zero  without  passing  it.  The  system 
is  then  just  aperiodic,  and  is  in  the  most  favorable  condition  for 
rapid  working.  Curve  IV  illustrates  this  case.  We  shall  desig- 
nate this  as  critical  aperiodic  motion.  If  B  is  still  larger,  that  is, 
if  the  damping  is  greater  than  the  critical  value,  the  return  to 
zero  is  slower.  As  B  increases  the  time  of  return  to  zero  increases 
more  and  more  and,  when  B  is  very  large,  the  galvanometer  is  in 
the  most  unfavorable  condition  for  rapid  working.  In  this  case 
the  zero  is  not  only  rendered  uncertain  but  the  movement  of  the 
system  is  very  slow  near  the  zero  position,  which  makes  it  uncer- 
tain when  the  zero  has  actually  been  reached.  The  same  remarks 
apply  to  the  outward  deflection  when  current  is  applied  to  the 
galvanometer,  the  outward  and  return  deflections  taking  place  in 
the  same  time  and  in  a  similar  manner.  Curves  V  and  VI  illus- 
trate this  condition. 

The  extent  to  which  a  galvanometer  system  is  damped  is  inde- 
pendent, in  the  case  of  moving-magnet  galvanometers,  of  the 


ART.  1504]  DEFLECTION   INSTRUMENTS  349 

resistance  in  the  circuit  external  to  the  galvanometer,  but  depends 
only  upon  the  frictional  or  air  damping  of  its  system.  In  this 
respect  a  moving-magnet  galvanometer  is  more  convenient  to  use 
than  a  moving-coil  galvanometer,  the  damping  of  which  varies 
greatly  with  the  resistance  in  its  external  circuit.  In  any  situa- 
tion or  with  any  type  of  galvanometer  the  best  conditions  of 
working  will  be  realized  when  the  damping  is  adjusted  so  the 
system  returns  to  zero  aperiodically,  as  exemplified  in  curve  IV. 
In  the  use  of  a  moving-magnet  galvanometer  this  adjustment  of 
the  damping  may  be  effected  by  shunting  the  galvanometer  with 
a  certain  resistance  or  by  hanging  upon  its  coil  a  closed  metallic 
circuit,  as  a  rectangle  of  copper  wire,  or  by  both  methods  combined. 

If  the  galvanometer,  of  the  moving  magnet  type,  is  used  in  low- 
resistance  measurements  with  a  Kelvin  double  bridge  it  will  always 
be  upon  a  closed  circuit  and  this  closed  circuit  may  have  a  resist- 
ance which  is  lower  than  the  critical  damping  resistance.  In  this 
case  the  galvanometer  will  be  overdamped  and  there  is  no  re- 
course for  reducing  the  damping  other  than  to  increase  the  torsion 
of  the  suspension,  or  to  weaken  the  strength  of  the  magnetic 
field;  but  both  of  these  procedures  will  diminish  the  sensibility 
of  the  galvanometer.  It  should  be  mentioned,  however,  that 
sensibility  is  directly  proportional  to  the  field  strength  while 
damping  is  proportional  to  the  square  of  the  field  strength,  and 
so  a  considerable  decrease  in  damping  may  be  effected  by  reducing 
the  field  strength  (by  magnetically  shunting  the  field  magnet) 
without  producing  a  corresponding  decrease  in  sensibility. 

1504.  Comparison  of  Galvanometers.  —  It  should  be  stated 
at  the  outset  that  "  sensibility  "  is  only  one  of  many  requisites 
which  a  galvanometer,  useful  for  resistance  measurements,  should 
possess.  Nor  will  an  understanding  and  solution  of  the  general 
equation  of  motion  of  a  galvanometer  system,  possessed  of  a 
moment  of  inertia,  a  moment  of  torsion  and  a  damping  factor,  be 
of  much  aid  either  in  the  design  or  in  the  selection  of  a  good 
galvanometer.  The  merits  of  any  galvanometer  should  be  judged 
by  its  adaptability  to  the  work  it  has  to  perform.  The  problem 
of  estimating  what  is  essential  to  usefulness  in  any  galvanometer 
must  be  solved  by  methods  which  only  experience  in  use  and 
design  can  teach. 

Every  galvanometer  design,  by  necessity,  is  a  series  of  com- 
promises. No  one  instrument  can  possess  in  full  measure  all 


350  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1504 

the  qualities  desired.  Experience  and  judgment  must  guide  to 
a  proportionate  selection  of  such  diverse  qualities  as  the  follow- 
ing: Sensibility,  short  period,  accurate  return  to  zero,  tensile 
strength  of  suspension,  freedom  from  easy  disturbance  by  external 
influences  such  as  air  currents,  accurate  balancing  of  the  system 
to  lessen  the  effect  of  vibrations,  proportionality  in  the  deflec- 
tions, strength  and  uniformity  of  the  magnetic  field,  clearness  of 
scale  and  ease  in  reading,  freedom  from  magnetic  impurities  in 
the  system  of  moving-coil  instruments,  variable  or  fixed  damp- 
ing to  secure  aperiodic  return  to  zero,  proper  coil  clearance  for 
free  motion  and  ease  of  adjustment,  arrangements  to  facilitate 
transportation,  freedom  from  parasitic  currents  resulting  from 
internal  thermoelectric  forces,  proper  resistance  of  coil  and  sus- 
pensions, permanence  of  the  galvanometer  constant  and  magnitude 
of  its  temperature  coefficient,  provisions  for  easy  inspection  of  sys- 
tem and  replacing  of  suspensions,  high  insulation  where  required, 
facility  in  mounting,  adaptability  to  general  classes  of  work  as  well 
as  to  a  specific  service,  ease  in  assembling,  grace  in  appearance,  and 
a  construction  which  is  moderate  in  cost.  Besides  understanding 
such  features  in  practical  design,  one  must  also  understand  the  fun- 
damental principles  which  underlie  every  galvanometer,  if  the  best 
possibilities  of  the  instrument  are  to  be  secured.  As  a  beginning 
in  design  or  in  selecting  an  instrument  for  use,  particular  qualities, 
required  for  a  particular  service,  should  have  special  attention. 
No  universally  serviceable  instrument  of  this  class,  equally  good 
for  all  kinds  of  work,  ever  was  or  ever  will  be  constructed. 

These  preliminary  remarks  hint  at  the  scope  of  the  subject  of 
galvanometer  construction,  and  emphasize  the  necessity  one  is 
under,  if  he  would  pass  judgment  upon  the  merits  of  a  galvanom- 
eter, to  have  first  a  clear  conception  of  the  use  for  which  the  in- 
strument is  intended.  A  galvanometer  admirably  adapted  to 
show  the  readings  of  a  bolometer  giving  the  energy  distribution 
in  the  spectrum  would  make  a  poor  lecture-room  instrument  for 
general  demonstration  purposes,  nor  would  it  be  suitable  to  put 
into  students'  hands  for  Wheatstone-bridge  work. 

A  common  basis  for  comparison  of  the  respective  worths  of 
galvanometers  of  diverse  pattern,  in  respect  to  the  single  quality 
which  has  been  termed  "  the  figure  of  merit/'  may  be  reached 
with  a  fair  degree  of  satisfaction.  A  clear  physical  conception 
of  this  feature  and  what  it  means  will  be  a  useful  preliminary 


ART.  1504] 


DEFLECTION   INSTRUMENTS 


351 


to  any  estimation   of   the  worth  of   any  current-measuring  in- 
strument. 

Let  (Fig.  1504)  Ga  and  Gb  represent,  diagramatically,  two  sys- 
tems of  any  type  of  moving-coil  galvanometer.  Let  these  systems 
be  held  by  suspensions  Sa  and  Sb  attached  at  points  pi,  p2  and 
pi  and  p2. 


FIG.  1504. 

Suppose  the  coil-winding  of  each  is  on  a  metal  frame  of  such 
cross-section  and  conductivity  that,  when  the  system  rotates  in 
the  magnetic  field,  its  return  to  zero  from  a  deflection  is  just 
aperiodic  in  virtue  of  the  currents  induced  in  the  frame.  To  have 
this  condition  always  fulfilled,  one  may  conceive  the  conductivity 
of  the  frame  to  vary  whenever  the  moment  of  inertia  of  the  sys- 
tem, the  torsion  of  the  suspension,  and  the  strength  of  the  mag- 
netic field  are  varied.  This  premised,  let  such  a  current  pass 
thru  each  system  that  it  will  be  rotated  thru  a  standard  angle 
6,  which  may  be  made  always  the  same  by  varying  the  strength 
of  the  current.  Evidently  the  current  which  will  be  required  to 
produce  this  deflection  will  depend  upon  many  factors,  chief 
among  which  are  the  strength  and  uniformity  of  the  magnetic 
field,  the  torsional  force  of  the  suspension,  the  length  and  number 
of  turns  in  the  coil,  and  the  degree  of  freedom  from  magnetic 
impurities  in  the  system. 

The  system,  starting  from  rest,  will  require  a  certain  time  T 
after  the  current  is  applied,  to  complete  a  certain  fractional  part 
of  its  aperiodic  deflection.  To  be  definite,  assume  that  the  de- 
flection is  practically  completed,  when  it  has  reached  within  0.05 
of  1  per  cent  of  its  final  deflection.  This  time  T  will  be,  for  all 


352  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1504 

purposes  of  practical  computation,  the  same  as  the  time  of  a 
complete  oscillation  of  the  same  system  undamped.  If  7  is  the 
moment  of  inertia  of  any  galvanometer  system,  then  the  square 
of  its  time  of  deflection,  as  above  denned,  or  of  a  complete  oscilla- 
tion if  undamped,  is  proportional  to  this  moment  of  inertia  —  or 


Now  so  arrange  matters  that  the  same  current  thru  each  of  the 
two  systems  Ga  and  Gb  will  produce  in  each  the  standard  angular 
deflection  b.  Then  the  sensibility  Sm  of  each  system  is  the  same 
and  this  sensibility  will  be  inversely  proportional  to  the  current 
required  to  produce  the  deflection.  If,  however,  the  system  Ga 
has  a  moment  of  inertia  Ia,  and  the  system  Gb  a  greater  moment  of 
inertia  /&,  then  Ga  will  reach  its  standard  deflection  in  a  shorter 
time  than  Gb.  The  ratio  will  maintain 


By  hypothesis,  the  sensibilities,  that  is,  the  currents  required 
thru  each  to  produce  the  standard  deflection,  are  equal.  Since, 
however,  Ga  deflects  in  a  shorter  time  than  Gb  its  suspension  might 
be  weakened  until  its  time  of  deflection  equals  that  of  Gb.  But 
with  a  weaker  suspension  it  will  take  less  current  to  produce 
the  standard  deflection.  Hence,  with  equal  times  to  make  the 
standard  deflection,  the  system  Ga  is  more  sensitive  than  the 
system  (7&. 

Conversely,  to  make  Gb  deflect  the  same  amount  in  the  same 
time  as  Ga,  its  suspension  must  be  stiffened,  and  with  a  stiffer 
suspension  it  will  require  more  current  than  Ga  for  the  standard 
deflection.  In  this  respect  it  is  a  less  sensitive  galvanometer 
than  Ga. 

As  it  is  always  possible  to  vary  within  wide  limits  the  torsional 
force  of  a  galvanometer  suspension,  a  galvanometer  which  is 
quick  but  not  sensitive  can  be  made  more  sensitive  at  the  expense 
of  quickness  by  changing  its  suspension,  and  a  galvanometer 
which  is  slow  but  sensitive  can  be  made  quicker  at  the  expense  of 
sensibility.  If  we  call  Sm  the  sensibility  of  a  particular  galva- 

nometer and  T  its  period,  then  the  product  Sm  ^  cannot  be  in- 
creased by  changes  of  the  above  character.     We  shall  call  this 


ART.  1504]  DEFLECTION  INSTRUMENTS  353 

product  proportional  to  the  useful  sensibility  of  any  particular 
galvanometer  and  write 

ff«f?-  a) 

By  T  we  must  understand  the  time  of  a  complete  oscillation, 
if  the  galvanometer  is  undamped,  or  the  time  it  takes  to  reach 
its  final  deflection  within  0.05  of  1  per  cent,  if  it  is  magnetically 
damped  to  be  just  aperiodic. 

For  practical  purposes  of  comparison  of  galvanometers,  the 
time  may  be  considered  the  same  for  the  instrument  in  either  of 
these  conditions. 

While  the  sensibility  Sm  is  inversely  proportional  to  the  current 
needed  for  the  standard  deflection,  this  current,  if  everything  else 
remains  the  same,  will  be  less  as  the  number  of  turns  in  the  coil 
is  increased.  Of  two  galvanometers  which  are  to  be  used  on  the 
same  constant-current  circuit,  and  which  are  alike  in  all  features 
except  in  respect  to  number  of  turns,  that  one  which  has  the  more 
turns  will  be  the  more  sensitive.  We  can  call,  therefore,  the 
sensibility  of  a  galvanometer  for  use  on  a  constant-current  cir- 
cuit a  quantity  which  is  proportional  to  its  number  of  turns  n, 
and  inversely  proportional  to  the  current  i  required  to  produce  a 
standard  deflection,  or 

&,«?.  (2) 

i 

Hence,  its  useful  sensibility  is 


To  increase  the  number  of  turns  we  may  proceed  in  either  or  both 
of  two  ways;  the  size  of  the  insulated  wire  may  be  diminished  and 
the  same  winding  space  be  filled,  or  the  wire  may  be  kept  the  same 
size  and  the  dimensions  of  the  cross-section  of  the  winding  chan- 
nel may  be  increased.  By  the  first  method  the  moment  of  inertia 
of  the  system  remains  nearly  the  same.  It  would  remain  exactly 
the  same,  if  in  altering  the  size  of  the  wire  no  alteration  were  made 
in  the  density  of  the  coil  by  changing  the  ratio  of  insulation  to 
wire,  thru  a  change  of  wire  size.  By  the  second  method,  the 
moment  of  inertia,  and  hence  T2,  will  be  changed  unless  the  length 
of  the  turns  are  also  diminished  in  a  proper  proportion.  We  have 
seen  that  U,  the  quantity  which  we  have  called  the  useful  sen- 


354  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1504 

sibility  of  a  galvanometer,  cannot  be  changed  by  changing  the 
torsional  force  of  its  suspension,  but  it  may  be  changed  by 
changing  the  coil  winding.  Thus,  in  changing  n,  if  the  moment 
of  inertia  only  is  changed  T2  will  be  changed  and  U  will  be  changed 
because  the  period  changes,  but  if  n  is  changed  in  such  a  manner 
as  not  to  change  the  period  then  U  will  change  again  because  Smj 
the  sensibility,  changes.  If  n  is  changed  in  such  a  way  as  to  vary 
both  Sm  and  T2,  U  will  still  change  unless  the  exceptional  condi- 

Tl 

tion  is  met,  that  n  so  changes  that  7™  remains  constant.     We  con- 

clude, by  the  above  line  of  reasoning,  that  useful  sensibility  is  a 
constant  property  of  a  particular  galvanometer  with  a  particular 
winding,  but  a  quantity  which  usually  varies  when  the  coil  windings 
are  changed.  But  if  we  divide  the  useful  sensibility  by  the  num- 
ber of  turns  in  the  coil  and  write 


we  obtain  the  new  quantity  F  which  has  been  designated  the 
"  figure  of  merit  "  of  a  galvanometer. 

The  figure  of  merit  of  a  galvanometer  is  a  kind  of  "  specific 
quantity  "  which  attaches  to  every  galvanometer.  As  the  use- 
ful sensibility  of  a  galvanometer  cannot  be  improved  by  changing 
the  torsional  force  of  its  suspension  so  also  we  cannot  increase  the 
figure  of  merit  of  a  galvanometer  by  changing  either  its  suspension 
or  the  turns  which  fill  a  winding  space  of  fixed  volume.  A  galva- 
nometer with  a  certain  "  figure  of  merit  "  is  potentially,  so  to 
speak,  capable  of  having  a  certain  chosen  period  with  a  certain 
accompanying  sensibility  and  number  of  turns,  or  a  certain  chosen 
sensibility  with  a  certain  accompanying  period  and  number  of  turns, 
but  to  increase  the  figure  of  merit  changes  must  be  made  in  the 
field  strength  or  in  the  proportioning  of  the  galvanometer  parts. 
As  "  figure  of  merit  "  attaches  as  a  specific  property  to  every 
galvanometer,  it  serves  in  a  useful  way  to  compare  the  intrinsic 
worths  of  various  types  of  instruments.  If,  however,  we  wish  to 
compare  the  figures  of  merit  of  different  galvanometers,  we  cannot 
do  so  practically  by  using  the  expression  above  in  its  present 
form,  because  there  is  no  easy  way  of  counting  the  number  of  turns 
in  the  coils  after  the  galvanometers  are  built.  It  is  necessary, 
therefore,  to  find  how  the  resistance  of  the  coil  is  related  to  its 


ART.  1504]  DEFLECTION   INSTRUMENTS  355 

number  of  turns,  for  this  is  a  quantity  easily  measured.     To  do 

this  write 

,  . 


where       Zi  =  length  of  mean  turn, 
W  =  diameter  of  wire  and 

p  =  specific  resistance  of  wire. 
If  S  =  cross-section  of  channel  and 

d  =  double  thickness  of  wire  insulation,  then 

S  TrS 


n  = 


(W  +  d)2      irW2+2wWd' 


(6) 


the  term  -K  d2  being  neglected,  as  being  very  small. 

From  Eq  (6)  we  derive  irW  2  =  —  —  —  ,  and  putting  this 

value  of  vW2  in  Eq.  (5)  we  obtain 


Solving  this  quadratic  [see  appendix  II,  7,  Eq.  (16)]  and  using  the 
positive  sign  before  the  radical,  we  obtain 

n  a  VRS  +  R2W2  d2  -  RW  d.  (8) 

When  this  value  of  n  replaces  n  in  Eq.  (4)  we  have  the  result- 
ing expression,  not  involving  n,  for  the  figure  of  merit  of  a 
galvanometer, 

p  _-  _  §5  _  /q\ 

T2[(RS  +  R2W2  d2)*  -  RW  d]  ' 


Except  in  cases  where  galvanometers  wound  with  coarse  wire 
are  compared  with  galvanometers  wound  with  very  fine  wire,  the 
thickness  d  of  the  insulation  may  be  neglected.  We  may  con- 
sider S  constant  and  we  have,  when  we  do  this, 

-^    acJ-  (10) 


T*VR 

S 

—  ^=.  is  the  usual  expression  for  the  figure  of  merit  of  any  galva- 
T2VR 

nometer.  In  comparing  galvanometers  by  it,  the  supposition  is 
made  that  thickness  of  insulation  is  neglected  and  that  the  gal- 
vanometers compared  are  wound  with  wire  of  the  same  specific 
resistance.  The  specification  which  we  shall  adopt  to  define  Sm 
is  as  follows: 


356  MEASURING  ELECTRICAL  RESISTANCE       [ART.  1504 

With  the  scale  at  1000  scale  divisions  from  the  mirror,  of  a 
mirror  galvanometer,  the  sensibility  Sm  is  the  number  of  megohms 
which  must  be  in  the  galvanometer  circuit  so  that  with  an  E.M.F. 
of  one  volt  in  the  circuit,  there  will  result  a  deflection  of  one  scale 
division.  With  this  understood,  we  can  write 


A  galvanometer  would  have,  then,  a  unit  figure  of  merit,  if  its 
time  of  a  complete  oscillation  is  one  second  (or  which  is  practically 
the  same  thing,  if  its  time  of  a  periodic  return  to  zero  within  0.05 
of  1  per  cent  of  its  previous  deflection  is  one  second),  and  the  re- 
sistance of  its  winding  is  one  ohm,  and  if,  with  one  megohm  in 
series  and  one  volt  in  circuit,  its  deflection,  on  a  scale  1000  scale 
divisions  from  its  mirror,  is  one  division. 

As  this  unit  has  received  no  name,  we  shall  call  it,  for  con- 
venience, a  D'  Arson. 

We  can  say,  also  in  accord  with  the  above  definition  of  sensi- 
bility, that  the  sensibility  is  unity  when  one  microampere  pro- 
duces the  standard  deflection.  Hence,  if  im  =  the  microamperes 
in  the  galvanometer  circuit, 


If  Em  microvolts  are  applied  at  the  terminals  of  the  galvanom- 
eter of  resistance  R,  we  have 

Em  0         R 

lm  =:  ~R      °r  =  E~  ' 

Putting  this  value  of  Sm  in  Eq.  (11)  gives 

'*-$'  (12) 

The  relation  (12)  defines  the  figure  of  merit  of  a  galvanometer 
in  terms  of  its  resistance,  period,  and  the  number  of  microvolts 
applied  at  its  terminals  to  produce  the  standard  deflection. 

As  an  example  of  the  use  of  relation  (11)  suppose  we  have  a 
galvanometer  with  a  complete  period  of  5  seconds,  a  coil  resistance 
of  400  ohms,  and  which  deflects  one  scale  division  with  one  volt 
acting  thru  500  megohms,  then  its  figure  o'f  merit  is 

500 
F=  --  7=  =  1  D'Arson. 

52A/400 


ART.  1504]  DEFLECTION   INSTRUMENTS  357 

If  this  galvanometer  were  given  a  longer  period,  by  using  a 
weaker  suspension,  its  sensibility  would  be  larger,  but  F  would 
not  be  altered  by  this  change.  It  is  evident  that  the  coil  might 
be  wound,  using  the  same  size  wire,  to  a  smaller  resistance,  but 
if  this  were  done  the  mass  of  the  coil  would  be  less  and  hence  T2 
would  be  smaller.  Both  of  these  changes  would  contribute  to  a 
greater  figure  of  merit.  On  the  other  hand,  a  smaller  resistance 
would  mean  a  smaller  number  of  turns  which  would  reduce  the 
sensibility  and  hence  diminish  the  figure  of  merit.  Thus  it  is 
always  open  to  the  designer  to  so  choose  the  winding  and  pro- 
portion the  coil  and  to  so  arrange  the  strength  of  the  magnetic 
field  and  other  factors  that  F  shall  be  large.  The  success  with 
which  he  does  this  determines  in  considerable  measure  the  per- 
fection of  his  design. 

It  must  not  be  forgotten,  however,  that  the  number  of  D'Arsons 
possessed  by  a  galvanometer  is  not  necessarily  a  final  measure  of 
its  fitness  for  actual  service.  It  may  possess  faults  of  many  kinds 
which  more  than  offset  a  large  figure  of  merit.  Chief  among  such 
is  zero  shift  and  magnetic  impurities  in  the  system,  which  two,  in 
fact,  generally  go  together.  There  are  also  other  common  defects, 
as  small  coil  clearance,  inaccessibility  of  the  parts,  a  poor  optical 
system,  an  unproportional  scale,  a  provoking  tendency  of  the 
system  to  respond  to  small  tremors,  and  a  host  of  other  minor 
defects  which  the  user  soon  observes  and  condemns. 

If  it  were  not  necessary  to  load  a  galvanometer  system  with  a 
mirror  or  a  pointer  for  the  purpose  of  reading  the  deflections,  it 
would  be  possible  by  proper  designing  and  by  a  great  diminution 
in  the  size  of  the  moving  parts  to  realize  an  instrument  which 
would  possess  an  enormous  figure  of  merit  as  compared  with  an 
ordinary  moving-coil  galvanometer.  This  has  in  fact  been  done 
in  the  case  of  the  Einthoven  String  Galvanometer,  which  has 
about  3000  times  the  number  of  D' Arsons  of  a  good  moving-coil 
galvanometer  using  a  mirror  and  scale.  We  are  led  thus  to  the 
general  consideration  of  what  are  the  possibilities  of  obtaining  a 
great  figure  of  merit  for  galvanometers  of  the  deflection  type. 

In  every  galvanometer  of  this  type  we  may  consider  the  moment 
of  inertia  of  its  moving  system  as  made  up  of  two  parts :  One  part 
is  the  moment  of  inertia  which  is  contributed  by  the  mirror,  the 
pointer,  or  whatever  device  may  be  attached  to  the  system  which 
is  required  for  reading  the  deflections  of  the  instrument.  We 


358  MEASURING  ELECTRICAL  RESISTANCE  [ART.  1504 

may  make  this  reading  device  small  but  we  cannot  dispense  with 
it  altogether  and  preserve  the  instrument  as  a  galvanometer  of  a 
type  to  which  the  name  is  ordinarily  applied.  Indeed,  there  are 
many  practical  considerations  which  soon  put  a  limitation  upon  a 
continual  diminution  of  "mass  of  these  parts.  The  other  part  is 
the  moment  of  inertia  which  belongs  to  the  moving  wire  or  mag- 
nets, which  constitute  the  effective  working  element  of  the  system. 
This  part  of  the  total  moment  of  inertia  can  be  modified  at  will 
by  the  designer  with  the  object  of  making  the  figure  of  merit  of 
the  galvanometer  as  large  as  possible.  The  question  then  arises, 
has  the  figure  of  merit  a  maximum  value  which  the  most  skilful 
designing  cannot  exceed?  If  there  were  no  "  dead  parts  "  attached 
to  the  system  for  reading  deflections,  then,  theoretically,  a  gal- 
vanometer could  be  given,  by  proper  designing,  an  indefinitely 
great  figure  of  merit.  This  realization  is  obtained  practically  in 
the  Einthoven  String  Galvanometer.  But  as  long  as  galvanom- 
eters continue  to  be  instruments  the  deflections  of  which  are  read 
with  mirrors  or  pointers,  there  will  be  a  theoretical  maximum 
figure  of  merit  which  cannot  be  exceeded.  The  author  has 
shown*  that  when  we  have  chosen  the  moment  of  inertia  of  the 
winding  of  a  rectangular  coil,  equal  to  the  moment  of  inertia  of 
the  "  dead  parts,"  mirror  or  pointer,  the  resistance  of  the  coil  re- 
maining always  the  same,  we  have  designed  the  proportions  of  the 
coil  such,  that  the  galvanometer,  in  this  respect,  has  the  greatest 
figure  of  merit  which  it  is  possible  to  give  it. 

Every  consideration  shows  that  in  starting  out  to  design  a 
galvanometer  of  any  type  which  is  to  have  a  large  number  of 
D'Arsons,  one  should  begin  by  carefully  considering  the  selection 
of  the  mirror,  pointer,  or  other  contrivance  essential  to  reading 
the  deflections,  so  that  this  contrivance  may  have  the  least  possible 
moment  of  inertia;  for  it  is  the  mass  of  these  "  dead  parts  "  which 
ultimately  sets  a  limit  to  the  maximum  figure  of  merit  obtainable. 
The  dispensing  of  "  reading  parts "  in  the  Einthoven  String 
Galvanometer  is  the  essential  reason  for  its  enormous  number  of 
D'Arsons.  Similar  considerations  hold  for  oscillographs,  watt- 
meters, pointer  voltmeters  and  ammeters  and  other  deflection 
instruments  in  which  a  high  figure  of  merit  is  desired. 

*  See  pages  255-256  Jour,  of  the  Franklin  Institute,  October,  1910, 
"The  Comparison  of  Galvanometers  and  a  New  Type  of  Flat-coil  Galvanom- 
eter." 


ART.  1504]  DEFLECTION  INSTRUMENTS  359 

In  the  above  discussion  attention  has  been  confined  chiefly 
to  galvanometers  intended  for  use  on  constant  current  or  nearly 
constant-current  circuits.  This  is  the  condition  which  applies 
when  galvanometers  are  used  for  measuring  insulation  resistance 
by  direct  deflection  methods.  It  also  applies,  tho  to  a  less  extent, 
when  galvanometers  are  used,  with  considerable  resistance  external 
to  themselves,  in  Wheatstone  and  Kelvin  double-bridge  measure- 
ments of  resistance.  When,  however,  galvanometers  are  de- 
signed for  use  with  thermocouples  and  for  reading  millivolt  or 
microvolt  drops  over  low  resistances,  many  points  of  design,  as 
the  galvanometer  resistance,  damping,  etc.,  should  receive  atten- 
tion; but  our  limits  will  not  permit  a  discussion  of  this  phase  of 
the  subject.  When  dealing  with  galvanometers  for  use  on  con- 
stant potential  circuits,  it  is  preferable  to  use,  as  the  expression 
for  the  figure  of  merit,  the  relation  (12)  given  above.  In  this 
expression  when  Em  =  I  microvolt,  T  =  1  second,  and  R  =  1  ohm, 
the  figure  of  merit  is  unity,  and  then  we  may  call  the  unit,  a 
microvolt  D' Arson. 

We  have  collected  in  a  table  the  essential  characteristics  of 
sixteen  different  well  known  types  of  galvanometers,  for  which 
see  table  at  end  of  paragraph. 

Number  16  is  a  galvanometer  of  the  D'Arsonval  type  made 
by  The  Leeds  and  Northrup  Company,  which  has  the  very 
high  figure  of  merit  of  2.80  D'Arsons.  Its  megohm  sensibility, 
however,  is  but  121.8.  Theoretically,  a  very  fine  suspension 
could  be  used  until  its  sensibility  reaches  that  of  No.  13  which  is 
1750  megohms.  Practically,  however,  this  would  not  be  feasible, 
because  the  suspension  would  be  finer  than  any  wire  on  the 
market  except  Wollaston  wire.  But  were  such  a  fine  suspension 
used,  the  more  serious  difficulty  would  arise  that  the  coil  would 
then  be  influenced  to  a  relatively  great  degree  by  traces  of  mag- 
netic matter  in  the  coil.  This  would  produce  a  large  zero  shift 
on  reversed  deflections.  Thus,  for  high  sensibility  work,  as 
in  cable  testing,  where  the  longer  period  is  not  too  serious  a  dis- 
advantage, galvanometer  No.  13  would  be  a  much  better  instru- 
ment to  use,  altho  its  figure  of  merit  is  but  11.4  per  cent  of 
that  of  No.  16. 

In  respect  to  Table  I  the  following  remarks  may  be  added. 
For  the  Einthoven  String  Galvanometer  (No.  1)  the  resistance  of 
the  string  was  reduced  to  its  copper  equivalent,  and  a  magnifica- 


360 


MEASURING  ELECTRICAL  RESISTANCE 


[ART.  1504 


tion  of  100  was  taken  as  the  equivalent  of  a  scale  at  1000  scale 
divisions  from  a  mirror. 

The  data  for  No.  2  was  taken  from  Siemens  and  Halske's  re- 
print No.  30,  and  the  data  for  No.  4  was  taken  from  an  article  by 
Dr.  H.  Sack  in  the  same  reprint.  The  data  for  No.  5  is  from 
The  Cambridge  Scientific  Instrument  Company's  catalogue.  The 
data  for  No.  16  is  based  on  the  average  of  five  instruments  designed 
by  the  author,  and  made  by  The  Leeds  and  Northrup  Company 
in  August,  1910. 

The  Weston  Voltmeter  (No.  7)  shows  a  figure  of  merit  of  0.115 
D'Arson.  In  considering  the  meaning  of  this  low  figure,  it  must 
be  remembered  that  a  light  system  has  to  carry  a  pointer  which 
must  be  heavy  enough  to  be  perfectly  rigid.  The  same  system, 
fitted  with  a  mirror,  would  show  a  much  higher  figure  of  merit. 
The  design  of  this  instrument,  as  every  one  knows,  is  most  scien- 
tifically worked  out,  and  the  fact  that  its  figure  of  merit  is  low 
simply  emphasizes  the  fact  that  one  must  exercise  great  caution 
against  estimating  the  real  worth  of  an  instrument  by  this  feature 
alone. 


No. 

Type  of  instrument 

Method  of  reading 

Critical 
resistance 
for  damp- 

Instru- 
ment 
resist- 
ance 

Megohm 
sensi- 
bility 

Com- 
plete 
period 

Figure  of 
merit  in 
D'  Arsons 

F-     Sm 

ing 

R 

Sm 

T 

T*VR 

*1 

Einthoven  string  

Microscope,  100  fold 

Aperiodic 

287 

5.2 

0.01 

3000.00 

magnification 

*2 

Dubois  Rubens,  iron- 

clad moving  mag- 

net   

Mirror 

Aperiodic 

290 

12200.0 

6.0 

20.00 

3 

Siemens    &    Halske, 

high  sensibility  .... 

Mirror 

120 

200 

2500.0 

12.0 

1.24 

*4 

Siemens    &    Halske, 

« 

high  sensibility...  . 

Mirror 

290 

1220  0 

6  0 

2  00 

*5 

Ayrton-Mather  

Mirror 

Undamped 

20 

52  6 

3.5 

0.96 

6 

R.    W.    Paul,    single 

pivot         millivolt- 

meter  

Pointer 

25 

50 

21.4 

3.0 

0.33 

7 

Weston,  voltmeter.  .  . 

Pointer 

Aperiodic 

75 

0.25 

0.5 

0.115 

8 

Weston,  portable  galv 

Pointer 

3000 

280 

13.8 

1.5 

0.37 

9 

L.  &N.,  typeP  

Mirror 

140 

124 

85.0 

8.5 

0.10 

10 

L.  &  N.,  marine  type 

Mirror 

Aperiodic 

1660 

21.4 

2.0 

0.131 

11 

L.  &    N.,    No.  2300 

standard  four  coil.  . 

Pointer 

300 

300 

15.54 

2.5 

0.14 

12 
13 

L.  &  N.,  typeH  
L.   &   N.,    No.    2280 

Mirror 

Aperiodic 

544 

295.0 

7.0 

0.26 

wide  coil  

Mirror 

Aperiodic 

1420 

1750.0 

12.0 

0.32 

14 

L.  &  N.,  steel  magnet 

Pointer 

500 

230 

8.8 

1.3 

0.344 

15 

L.  &  N.,  special  small 

four  coil  

Mirror 

500 

85 

6.4 

1.0 

0.695 

16 

L.&N.,No.2280nar- 

row  coil 

Mirror 

200 

180 

121.8 

1.8 

2.80 

Data  not  obtained  by  author. 


ART.  15051 


DEFLECTION   INSTRUMENTS 


361 


FIG.  1505. 

1505.  Description  of  One  Type  of  High-sensibility  Galvanom- 
eter. —  As  it  is  not  our  purpose  to  discuss  the  structural  details 
of  galvanometers  as  made  by  the  various  instrument  makers  we 
can  do  no  more  in  this  respect  than  briefly  describe,  and  give 
an  illustration  of,  a  single  type  of  high-sensibility  galvanometer 


362  MEASURING  ELECTRICAL  RESISTANCE          [ART.  1505 

(galvanometer  2280;  Nos.  13  and  16  in  the  above  table).  The 
instrument  is  shown  in  Fig.  1505.  It  is  furnished  with  a  coil  of 
medium  width  when  intended  for  use  in  insulation  testing  and 
where  high  sensibility  is  more  important  than  a  quick  working 
period.  It  may  also  be  furnished  with  a  very  narrow  coil,  when 
it  becomes  best  adapted  to  Wheatstone-bridge  work,  low  resist- 
ance measurements  by  the  Kelvin  double-bridge  principle  and 
general  laboratory  work  in  connection  with  potentiometers,  ther- 
mocouples, etc. 

Its  more  prominent  features  may  be  summarized  as  follows : 

The  tube  which  contains  the  coil  system  makes  one  unit  and  the 
magnet  another,  so  different  tubes  with  coils  of  different  charac- 
teristics may  be  fitted  to  the  same  magnet. 

The  coil  system  is  always  exposed  to  full  view. 

The  deflections  are  very  closely  proportional  to  the  current 
passed  thru  the  coil. 

The  damping  of  the  system  may  be  varied  thru  wide  limits  by 
removable  copper  rectangles  which  fit  upon  the  coil. 

The  suspended  system  may  be  locked  by  a  clamping  device 
when  the  instrument  is  to  be  carried  about. 

The  instrument  as  a  whole  is  very  highly  insulated  from  ground 
by  hard  rubber,  petticoat-insulated  leveling  screws. 

The  figure  of  merit  is  high,  the  narrow  coil  type  reaching  2.8 
D'Arsons,  while  the  wider  coil  type  of  1500  ohms  resistance  has  a 
megohm  sensibility  of  1700  megohms  with  a  period  of  12  seconds. 

The  suspension  is  easily  replaced  if  broken,  an  event  which  is 
made  unlikely  by  means  of  a  protecting  spring  at  the  top  of  the 
suspension  tube. 


APPENDIX 


I.  — TABLE. 

a 


(1)  Values  of 


1000  -a 


U 

lits 

100 

10 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

0 

.00 

0000 

1001 

2004 

3010 

4016 

5025 

6036 

7049 

8064 

9082 

1 

.0 

1010 

1112 

1214 

1317 

1420 

1523 

1626 

1730 

1833 

1937 

2 

.0 

2011 

2145 

2250 

2354 

2459 

2564 

2670 

2775 

2881 

2987 

3 

.0 

3093 

3199 

3306 

3413 

3520 

3627 

3735 

3843 

3950 

4058 

4 

.0 

4167 

4275 

4384 

4493 

4602 

4712 

4820 

4931 

5042 

5152 

5 

.0 

5263 

5374 

5485 

5596 

5708 

5820 

5932 

6044 

6156 

6269 

6 

.0 

6383 

6496 

6610 

6724 

6838 

6952 

7066 

7180 

7296 

7412 

7 

.0 

7527 

7643 

7759 

7875 

7992 

8109 

8225 

8342 

8460 

8578 

8 

.0 

8696 

8814 

8933 

9051 

9170 

9290 

9408 

9528 

9649 

9770 

9 

.0 

9890 

1001 

.1013 

.1025 

.1037 

.1050 

.1062 

.1074 

.1086 

.1099 

0 

.1111 

1123 

.1136 

.1148 

.1160 

1173 

.1186 

.1198 

.1211 

.1223 

1 

.1236 

1248 

.1261 

.1274 

.1287 

.1300 

.1312 

.1325 

.1338 

.1351 

2 

.1364 

1377 

.1390 

.1403 

1416 

.1429 

.1442 

.1455 

.1468 

.1481 

3 

.1494 

1507 

.1521 

.1534 

1547 

.1561 

.1574 

.1587 

.1601 

1615 

4 

1628 

1641 

.1655 

.1669 

.1682 

.1695 

.1710 

.1723 

.1737 

.1751 

5 

.1765 

1778 

.1792 

J806 

.1821 

.1834 

.1848 

.1862 

.1876 

.1890 

6 

.1905 

1919 

.1933 

.1947 

.1962 

.1976 

.1990 

.2005 

.2019 

.2034 

7 

2048 

2063 

.2077 

.2092 

.2106 

.2121 

.2136 

2151 

.2165 

.2180 

1 

8 

.2195 

.2210 

.2225 

.2240 

.2255 

2270 

.2285 

.2300 

.2315 

.2331 

1 

9 

.2346 

.2361 

.2376 

.2392 

.2407 

.2423 

.2438 

.2454 

.2469 

.2485 

2 

0 

.2500 

2516 

.2532 

.2547 

.2563 

.2579 

.2595 

2610 

.2625 

.2642 

2 

1 

.2658 

.2674 

.2630 

.2706 

.2722 

.2739 

.2755 

.2772 

.2788 

.2804 

2 

2 

.2820 

.2837 

.2853 

.2870 

.2887 

2903 

.2920 

.2937 

.2954 

.2971 

2 

3 

.2987 

.3004 

.3020 

.3038 

.3055 

3072 

.3089 

3106 

.3123 

.3140 

2 

4 

3157 

.3175 

.3192 

.3210 

.3228 

.3245 

.3262 

.3280 

.3298 

.3316 

2 

5 

.3333 

.3351 

.3369 

3387 

.3405 

3423 

.3440 

.3459 

.3477 

.3495 

2 

6 

.3513 

.3532 

.3550 

.3568 

.3587 

3606 

.3624 

3643 

.3662 

.3681 

2 

7 

.3699 

.3717 

3736 

.3755 

.3774 

3793 

.3812 

.3831 

.3850 

.3869 

2 

8 

.3889 

.3908 

3928 

.3947 

.3966 

3986 

.4005 

.4024 

.4044 

.4064 

2 

9 

.4084 

.4104 

.4124 

.4144 

.4164 

.4185 

.4205 

.4225 

.4245 

.4265 

3 

0 

.4285 

.4306 

.4326 

.4347 

.4368 

4389 

.4409 

4430 

.4450 

.4471 

3 

1 

.4493 

.4514 

.4535 

.4556 

.4577 

4598 

.4619 

4640 

.4661 

.4683 

3 

2 

4705 

.4727 

.4749 

.4771 

.4793 

4814 

4836 

.4858 

.4881 

.4903 

3 

3 

.4925 

.4947 

.4969 

.4992 

.5015 

.5038 

.5060 

.5083 

.5106 

.5129 

3 

4 

.5152 

.5174 

5197 

.5220 

.5244 

.5267 

.5290 

.5313 

.5336 

.5360 

3 

5 

5384 

.5407 

.5431 

.5455 

.5480 

5504 

.5528 

5553 

.5576 

.5600 

3 

6 

.5625 

.5650 

5674 

.5698 

.5723 

.5748 

.5773 

5798 

.5823 

.5848 

3 

7 

.5873 

.5899 

5924 

.5949 

.5974 

.6000 

.6025 

6051 

.6077 

.6103 

3 

8 

6129 

.6155 

6181 

.6207 

.6233 

6260 

.6286 

.6313 

.6340 

.6367 

3 

9 

.6394 

.6420 

.6447 

.6474 

.6502 

.6529 

.6557 

.6584 

.6611 

.6638 

4 

0 

.6666 

.6694 

.6722 

.6750 

.6778 

.6806 

.6834 

6862 

.6891 

.6920 

4 

1 

6949 

.6978 

7007 

.7036 

.7065 

.7094 

.7123 

.7152 

.7181 

.7211 

4 

2 

.7241 

.7271 

.7301 

.7331 

.  7361 

.7391 

.7421 

.7451 

.7482 

.7512 

4 

3 

.7543 

.7574 

7605 

.7636 

.7667 

.7698 

.7729 

.7760 

.7792 

.7824 

4 

4 

7857 

.7889 

.7921 

.7953 

.7986 

.8018 

.8050 

.8084 

.8117 

.8150 

4 

5 

8182 

.8215 

8248 

.8282 

.8316 

.8349 

.8382 

.8416 

.8450 

.8484 

4 

6 

.8518 

.8552 

8586 

.8620 

.8655 

.8691 

.8727 

.8762 

.8798 

.8834 

4 

7 

8868 

.8904 

8939 

.8975 

.9011 

.9048 

.9084 

.9120 

.9157 

.9194 

4 

8 

9231 

.9267 

9304 

.9341 

.9379 

.9417 

.9454 

.9493 

.9531 

.9570 

4 

9 

9609 

.9649 

9687 

.9725 

.9764 

.9803 

.9842 

.9881 

.9920 

.9960 

363 


364 


APPENDIX 


Values  of 


1000 -a 


Units 

100 

10 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

5 

0 

1.000 

1.004 

1.008 

1.012 

1.016 

.020 

1.024 

1.028 

.032 

.036 

5 

1 

1.041 

1.045 

1.049 

1.053 

1.058 

.062 

1.066 

1.071 

.075 

.079 

5 

2 

1.083 

1.088 

1.092 

1.097 

1.101 

.105 

1.110 

1.114 

.119 

123 

5 

3 

1.128 

1.132 

1.137 

1.141 

1.146 

.15 

1.155 

1.160 

.165 

.169 

5 

4 

1.174 

1.179 

1.183 

1.188 

1.193 

.198 

1.203 

1.208 

.212 

.217 

5 

5 

1.222 

1.227 

1.232 

1.237 

1.242 

.247 

1.252 

1.257 

.262 

.267 

5 

6 

1.273 

1.278 

1.283 

1.288 

1.294 

.298 

1.304 

1.309 

.314 

320 

5 

7 

1.326 

1.331 

1.336 

1.342 

1.347 

.353 

1.359 

1.364 

.370 

1.375 

5 

8 

.381 

1.386 

1.392 

1.398 

1.404 

.410 

1.415 

1.421 

.427 

1.433 

5 

9 

.439 

1.445 

1.451 

1.457 

1.463 

.469 

1.475 

1.481 

.487 

1.494 

6 

0 

.500 

1  506 

1.513 

1.519 

1.525 

.53 

1.538 

1.544 

1.551 

1.557 

6 

1 

.564 

1.571 

1.577 

1.584 

1.591 

.597 

1.604 

1.611 

1.618 

1.625 

6 

2 

.632 

1.639 

1.645 

1.652 

1.659 

.667 

1.674 

1.681 

1.688 

1.695 

6 

3 

.703 

1.710 

1.717 

1.724 

1.732 

.740 

1.747 

1.755 

1.763 

1.770 

6 

4 

.778 

1.786 

1.793 

1.801 

1.809 

.817 

1.825 

1.833 

1.841 

1.849 

6 

5 

.857 

1.865 

1.873 

1.882 

1.890 

.899 

1.907 

1.916 

1.924 

1.933 

6 

6 

.941 

1.950 

1.958 

1.967 

1.976 

.985 

1.994 

2.003 

2.012 

2.021 

6 

7 

2.030 

2.039 

2.048 

2.058 

2.068 

2.078 

2.087 

2.096 

2.106 

2.115 

6 

8 

2.125 

2.135 

2.145 

2.155 

2.165 

2.175 

2.185 

2.195 

2.205 

2.215 

6 

9 

2.225 

2.236 

2.247 

2.257 

2.268 

2.278 

2.289 

2.300 

2.311 

2.322 

7 

0 

2.333 

2.344 

2.355 

2.367 

2.378 

2.389 

2.401 

2.413 

2.425 

2.436 

7 

1 

2.448 

2.460 

2.472 

2.485 

2.497 

2.509 

2.521 

2.534 

2.546 

2.559 

7 

2 

2.571 

2.584 

2.597 

2.610 

2.623 

2.636 

2.650 

2.663 

2.676 

2.690 

7 

3 

2.703 

2.716 

2.731 

2.745 

2.759 

2.774 

2.788 

2.802 

2.817 

2.831 

7 

4 

2.846 

2.861 

2.876 

2.891 

2.907 

2.922 

2.937 

2.953 

2.968 

2.984 

7 

5 

3.000 

3.016 

3.032 

3.049 

3.065 

3.081 

3.098 

3.115 

3.132 

3.150 

7 

6 

3.168 

3.185 

3.202 

3.220 

3.237 

3.255 

3.273 

3.291 

3.310 

3.329 

7 

7 

3.348 

3.367 

3.386 

3.405 

3.425 

3.445 

3.464 

3.484 

3.505 

3.525 

7 

8 

3.546 

3.566 

3.587 

3.608 

3.630 

3.652 

3.674 

3.695 

3.717 

3.740 

7 

9 

3.762 

3.785 

3.808 

3.831 

3.854 

3.878 

3.902 

3.926 

3.950 

3  975 

8 

0 

4.000 

4.025 

4.050 

4.075 

4.102 

4.127 

4.154 

4.181 

4.209 

4.236 

8 

1 

4.263 

4.290 

4.319 

4.348 

4.376 

4.405 

4.435 

4.464 

4.494 

4.525 

8 

2 

4.556 

4.587 

4.618 

4.650 

4.682 

4.715 

4.748 

4.781 

4.814 

4.848 

8 

3 

4.882 

4.917 

4.953 

4.988 

5.025 

5.061 

5.097 

5.135 

5.173 

5.211 

8 

4 

5.250 

5.290 

5.330 

5.370 

5.411 

5.451 

5.493 

5.536 

5.580 

5.623 

8 

5 

5.666 

5.711 

5.757 

5.803 

5.850 

5.898 

5.945 

5.994 

6.043 

6.093 

8 

6 

6.143 

6.194 

6.247 

6.300 

6.353 

6.407 

6.463 

6.519 

6.576 

6.634 

8 

7 

6.693 

6.752 

6.812 

6.873 

6.937 

7.000 

7.064| 

7.129 

7.196 

7.264 

8 

8 

7.334 

7.403 

7.474 

7.546 

7.620 

7.696 

7.772 

7.849 

7.928 

8.009 

8 

9 

8.091 

8.175 

8.259 

8.346 

8.434 

8.524 

8.616 

8.709 

8.804 

8.901 

9 

0 

9.000 

9.101 

9.204 

9.309 

9.416 

9.526 

9.638 

9.753 

9.870 

9.989 

9 

1 

10.11 

10.23 

10.36 

10.49 

10.63 

10.76 

10.90 

11.05 

11.19 

11.34 

9 

2 

11.50 

11.66 

11.82 

11.99 

12.16 

12.33 

12.51 

12.70 

12.89 

13.08 

9 

3 

13.28 

13.49 

13  71 

13.93 

14.15 

14.38 

14.62 

14.87 

15.13 

15.40 

9 

4 

15.66 

15.95 

16.24 

16.54 

16.86 

17.18 

17.52 

17.87 

18.23 

18.61 

9 

5 

19.00 

19.41 

19.83 

20.28 

20.75 

21.22 

21.73 

22.26 

22.81 

23.38 

9 

6 

24.00 

24.64 

25.32 

26.03 

26.77 

27.57 

28.41 

29.30 

30.25 

31.26 

9 

7 

32.33 

33.49 

34.70 

36.04 

37.46 

39.00 

40.67 

42.48 

44.44 

46.62 

9 

8 

49.00 

51.63 

54.55 

57.83 

61.50 

65.67 

70.43 

75.93 

82.33 

89.91 

9 

9 

99.00 

10.1 

24.0 

41.9 

65.7 

99.0 

249.0 

332.3 

99.0 

99.0 

MATHEMATICAL  QUANTITIES  AND  RELATIONS        365 

II.  — MATHEMATICAL  QUANTITIES  AND  RELATIONS. 

(1)  Functions  of  TT  and  e. 

TT  is  the  ratio  of  the  circumference  to  the  diameter  of  a  circle. 
e  is  the  basis  of  the  natural  logarithms. 
TT  =  3.14159 

-  =  0.31830 

7T 

7T2  =  9.86960 
VT  =  1.77245 
logio  TT  =  0.4971498 
loge  TT  =  1.1447298 

e  =  2.71828 

-  =  0.36788 
e 

logio  e  =  0.4342944 
loge  e  =  1 

loge  10  =  2.3025850 
logio  x  =  0.43429  loge  x. 
loge  x  =  2.30258  logio  x. 
If  x  =  ev,  then  y  =  loge  x. 

(2)  English-Metric  and  Metric-English  Conversions. 
1  inch  =  2.54001  centimeters. 

1  centimeter  =  0.3937  inch. 

1  foot  =  0.304801  meter. 

1  meter  =  3.28083  feet. 

1  mile  =  1.60935  kilometers. 

1  kilometer  =  0.62137  mile. 

1  square  inch  =  6.452  square  centimeters. 

1  square  centimeter  =  0.1550  square  inch. 

1  cubic  inch  =  16.3872  cubic  centimeters. 

1  cubic  centimeter  =  0.0610  cubic  inch. 

1  U.  S.  liquid  gallon  =  3.78543  liters  =  231  cubic  inches. 

1  liter  =  0.26417  U.  S.  liquid  gallon. 

1  avoirdupois  pound  =  0.45359  kilogram. 
1  kilogram  =  2.20462  avoirdupois  pounds. 
1  avoirdupois  ounce  =  28.3495  grams. 
1  gram  =  0.03527  avoirdupois  ounce. 


366  APPENDIX 

(3)  Formulce  for  the  Conversion  of  Temperature  Scales. 

If  F  =  Fahrenheit  degrees  and  C  =  Centigrade  degrees,  then 
C  =  f(F  -  32).  0°  C.  =  +  32°  F. 

F  =  |  C  +  32.  0°  F.  =  -  17.777°  C. 

20°  C.  =  +  68°  F. 
-  40°  C.  =  -  40°  F. 
100°  C.  =  212°  F. 

(4)  Formulce  for  Temperature  Coefficients. 

If  a  linear  relation  exists  between  electrical  resistance  (or  re- 
sistivity) and  temperature  (only  true,  usually,  to  a  first  approxi- 
mation and  for  short  ranges  of  temperature)  then  these  relations 
hold, 

Pt  =  PO  (1  +  «0,  (1) 

Ptl  =  PO  (1  +  cdd,  (2) 

Pt,      —     Pt        '  ,QN 

«  =  T-   —j—>  (3) 

hpt  —  tpti 

Pt  —  Po 

a  =  — ,  (4) 

tpo 

where  p0  =  resistance  (or  resistivity)  at  temperature  0  degrees, 
pt  —  resistance  (or  resistivity)  at  temperature  t  degrees, 
ptl  =  resistance  (or  resistivity)  at  temperature  t\  degrees, 
and        a  =  temperature-resistance   coefficient  which,  by  the  as- 
sumption made,  is  a  constant  quantity. 

If  a  linear  relation  between  resistance  and  temperature  does  not 
hold,  then  the  resistance  (or  resistivity)  at  temperature  t  can  be 
expressed,  sometimes,  in  terms  of  the  resistance  (or  resistivity)  at 
0  degrees  by  the  relation 

Pt  =  PO  (1  +  at  +  fa2),  (5) 

where  a  and  b  are  constants  which  can  be  determined  by  experi- 
ment. 
Let  <rt  =  o-o  (1  -  00  (6) 

express  the  variation  of  electrical  conductivity  with  temperature, 
upon  the  assumption  that  relation  (1)  above  holds,  then 


and 

0 

<*  =  T= 


MATHEMATICAL  QUANTITIES  AND   RELATIONS         367 

(5)  Relations   between   Resistance   and    Conductivity.     (Consult 
Chapter  VII.) 

_  0.0203  W  .  n 

L*  —  R^r  30> 

0.01594  Z0 
or  C8=    -g-g-     30. 

C8  =  per  cent  conductivity  by  Matthiessen's  meter-millimeter 

standard. 

I  =  length  of  conductor  in  meters. 

d  =  diameter  in  millimeters  of  conductor  of  uniform  and  cir- 
cular cross-section. 

5  =  cross-section  in  square  millimeters  of  conductor  of  uniform 

cross-section. 

Rt  =  resistance  in  ohms  of  a  length  I  meters  at  temperature 
t  degrees  C. 

6  =  —  =  a  temperature  coefficient,   or  the  ratio  of   the  re- 

Po 

sistivity  of  the  material  at  temperature  t  degrees  C.  to  the 
resistivity  of  the  material  at  0  degrees  C. 
When  the  temperature  is  20°  C.  d  =  1.07968  for  copper. 
Use  for  calculation  the  value  6  =  1.08. 
It  is  nearly  the  same  for  all  pure  metals. 

If  the  cross-section  is  given  in  circular  mils  and  is  called  V  and  we 
write  y 

S  =  1973^2' 
then 


. 
C,=  lOO,  (4) 

and  17Q  72/j 


Cw  —  per    cent    conductivity    by    Matthiessen's    meter-gram 

standard. 
W  =  weight  of  I  meters  expressed  in  grams. 

C.'  =  ^C.',.          .  (6) 

where  Cwr  and  C,'  stand  for  conductivities  instead  of  per  cent 
conductivities  and 

5  =  density  of  the  material  of  the  conductor. 


368  APPENDIX 

Example  (1). 

A  copper  wire  is  2  mm  in  diameter  and  4  m  long,  and  has  a 
resistance  at  20°  C.  of  0.0223  ohm.  By  Eq.  (1)  its  per  cent  con- 
ductivity by  Matthiessen's  meter-millimeter  standard  is 

C,  =  --  Q  Q223  X  4     ~  10°  =  9^'^6  per  cent  conductivity. 

Example  (2). 

An  aluminum  wire  is  2  m  long  and  weighs  50  grams.  Its  re- 
sistance at  0°  C.  is  0.005585  ohm.  Then  6  =  1  and  by  Eq.  (5)  its 
per  cent  conductivity  by  Matthiessen's  meter-gram  standard  is 

0  14173  X  4  X  1 

=  2°3    er  Cent 


"        0.005585X50 

Example  (3). 

The  density  of  aluminum  is  2.7  and  its  conductivity  is  0.615 
by  the  meter-millimeter  standard,  then  by  Eq.  (6)  its  conductivity 
by  the  meter-gram  standard  is 

Q    OQ 

Cu'  =  ^  X  0.615  =  2.0249  conductivity. 

For  bimetallic  conductors,  as  copper  or  aluminum  conductors, 
with  a  steel  core  we  have  the  following  relations: 


. 

A3    ~|~   S 

S  =  cross-section  of  the  outside  metal  and 

s  =  cross-section  of  core  of  the  bimetallic  conductor. 


(7) 


S  and  s  may  be  expressed  in  terms  of  any  convenient  unit  pro- 
vided the  same  unit  is  chosen  for  each. 

C8"  =  conductivity  of  the  outside  metal  and 
C/  =  conductivity  of  the  metal  of  the  core,  both  by  the  meter- 
millimeter  standard. 

Example  (4). 

A  conductor  is  made  up  of  a  steel  core  which  has  a  con- 
ductivity, by  the  meter-millimeter  standard,  0.16,  and  of  an 
aluminum  covering  which  has  a  conductivity  0.61.  The  cross- 
section  of  the  core  is  2000  circular  mils  and  that  of  the  out- 
side metal  is  4000  circular  mils,  Then  by  Eq.  (7)  the  per  cent 


MATHEMATICAL  QUANTITIES  AND  RELATIONS        369 

conductivity  by  the  meter-millimeter  standard  of  the  bimetallic 
conductor  is 

4000  X  0.61  +  2000  X  0.16  inn  ,.  ., 

C8  =  -  40QO  +  2000  --         =       Per  C6     conductlvlty- 

r     TFiO/'  +  wv  ,. 

Cw=        Wi  +  W* 

Wi  =  the  weight  of  the  outside  metal  and 
W2  =  the  weight  of  the  core. 

These  weights  may  be  expressed  in  any  unit,  provided  the  same 
unit  is  used  for  both. 
Cw"  =  conductivity  of  outside  metal  and 
CJ  =  conductivity  of  core;  both  by  the  meter-gram  standard. 

Example  (5). 

A  conductor  has  a  steel  core  of  conductivity  0.20  by  the  meter- 
gram  standard,  and  an  aluminum  covering  of  conductivity  2  by 
the  same  standard. 

For  a  given  length  of  the  conductor  the  steel  core  weighs  10  Ibs., 
and  the  aluminum  covering  8  Ibs.  Then  by  Eq.  (8) 

Cw  =  8X2+10X0.20  100  =  100  per  cent  conductivity. 
o  +  lu 

"  '  nn  xqv 


8.89  (q  +  1) 

q  =  ratio    of   the   cross-section   of   the   outside   metal   to   the 

cross-section  of  the  core. 
62  =  density  of  the  outside  metal. 
61  =  density  of  the  metal  in  the  core. 
Cwf  and  Cw"  have  the  same  meaning  as  in  Eq.  (8). 

Example  (6). 

In  the  conductor  under  example  (5)  the  density  5i  of  the  steel 
core  is  7,  and  the  density  <52  of  the  aluminum  covering  is  2.7. 
The  ratio  of  the  cross-section  of  the  outside  metal  to  that  of  the 
core  is,  for  the  case  chosen, 


As  in  example  (5)  taking  CJ  =  0.2  and  Cw"  =  2  then  by  Eq.  (9) 
the  per  cent  conductivity  by  the  meter-millimeter  standard  is 


370  APPENDIX 

2.074  X  2.7  X  2  +  7  X  0.2 
C8  =          8  89  X  (2  074  +  1)          :  per  C       conductivity. 

0.01594  Id" 
R*  =  SCs»  +  sCa>' 

-p 

6"  =  -FT  =  the  temperature  coefficient  of  the  bimetallic  conductor. 

/to 

Rt  =  ohmic  resistance  in  ohms  at  t  degrees  centigrade  of  I  meters 

of  the  bimetallic  conductor. 
Other  quantities  have  the  same  meaning  as  above. 


Example  (7). 

A  bimetallic  conductor  304.8  m  long  has  a  steel  core  16.72  sq. 
mm  cross-section  and  a  copper  covering  25.59  sq.  mm.  cross- 
section.  The  conductivity  by  the  meter-millimeter  standard  of 
the  core  is  0.16,  and  of  the  covering  1.00.  If  at  20°  C.  the  coeffi- 
cient 0"  is  found  to  equal  1.09,  then  by  Eq.  (10)  the  resistance  of 
this  conductor  at  20°  C.  is 

0.01594  X  304.8  X  1.09 
R™  "  25.59  X  1  +  16.72  X  0.16  = 

0.14173  M" 
Cw"Wi  +  CV'W* 

Example  (8). 

A  bimetallic  conductor  is  1000  m  long.  The  weight  Wi  of  its 
steel  core  is  180,000  grams,  and  the  weight  of  its  aluminum  covering 
is  59,100  grams.  The  conductivity  of  the  core  by  the  meter-gram 
standard  is  0.2,  and  of  the  covering  2.03.  At  20°  C.  its  tempera- 
ture coefficient  is  found  to  be  1.09.  Then  by  Eq.  (11)  its  resist- 
ance is 

„  0.14173  X  10002  X  1.09 


2.03  X  59100  +  0.2 


, 
= 


$  =  value  in  dollars  of  a  certain  length  of  a  certain  conductor 
which  weighs  W  Ibs.,  and  has  a  conductivity  Cw  by  the  meter- 
gram  standard,  if  $1  =  the  value  in  dollars  of  the  same  length  of 
100  per  cent  conductivity  (by  the  meter-gram  standard)  conductor 
which  weighs  W\  Ibs. 


MATHEMATICAL  QUANTITIES  AND  RELATIONS        371 

Example  (9). 

Suppose  that  100  Ibs.  of  a  certain  length  of  100  per  cent  con- 
ductivity copper  is  worth  $20,  then  by  Eq.  (12)  250  Ibs.  of  the 
same  length  of  aluminum  wire  of  2.03  conductivity  or  203  per  cent 
conductivity,  by  the  meter-gram  standard,  is  worth 
_  20  X  250  X  2.03  _ 

"loo~ 

This  is  the  value  of  the  aluminum  as  compared  with  the  copper 
wire  when  considered  only  as  a  conveyer  of  electric  power.  Equal 
weights  of  the  same  length  of  aluminum  and  copper  have  values, 
as  conveyers  of  electric  power,  in  the  ratio  of  about  2  to  1. 

$'  =  ^sc8.  (is) 

Oi 

$'  =  the  value  in  dollars  of  a  certain  length  of  a  conductor  which 
has  a  cross-section  S  and  a  conductivity  C8  by  the  meter-millimeter 
standard,  when  $n  =  the  value  in  dollars  of  the  same  length  of 
100  per  cent  conductivity  conductor  (by  the  meter-millimeter 
standard)  which  has  a  cross-section  Si. 

Example  (10). 

Suppose  that  100  feet  of  copper  conductor  of  105,625  circular 
mils  is  worth  $64,  then  by  Eq.  (13)  100  feet  of  aluminum  conductor 
of  conductivity  0.6,  by  the  meter-millimeter  standard,  and  133,225 
circular  mils  is  worth 
64 
'  105625  X  133225  X  °'6  =  |48-44' 

Equal  cross-sections  of  the  same  lengths  of  aluminum  and  copper 
have  values  as  conveyers  of  electric  power  in  the  ratio  of  about 
6  to  10. 

(6)  Conversions  from  Practical,  to  Electrostatic,  to  Electromagnetic 
Units.  (C.G.S.  System.) 

v  =  velocity  of  light  =  3  X  1010  centimeters  per  second. 

N  =  any  number.  Elst  =  electrostatic.  Elmg  =  electromag- 
netic. 

N  coulombs  =  N  3  X  109  Elst  units  =  N  1Q-1  Elmg  units. 

1  Elst  unit  of  electricity  =  v~l  Elmg  unit. 

N  volts  =  ^  X  lO-2  Elst  units  =  N  108  Elmg  units. 


372  APPENDIX 

1  Elst  unit  of  potential  difference  =  v  Elmg  units. 

N  amperes  =  N  3  X  109  Elst  units  =  N  lO'1  Elmg  units. 

1  Elst  unit  of  electric  current  =  v~l  Elmg  unit. 

N  ohms  =  ^  X  10-11  Elst  units  =  N  109  Elmg  units. 

1  Elst  unit  of  resistance  =  v2  Elmg  units. 
N  farads  =  N  9  X  1011  Elst  units  =  N  1Q-9  Elmg  units. 
N  microfarads  =  N  9  X  105  Elst  units  =  N  10~15  Elmg  units. 
1  Elst  unit  of  capacity  =  v~2  Elmg  unit. 

TV 
TV  henrys  =  -5-  X  10~u  Elst  units  =  N  109  Elmg  units. 

1  Elst  unit  of  self-induction  =  v2  Elmg  units. 

25 

Examples :  25  ohms  =  -=-  X  10-uElst  units  =  25  X 109  Elmg  units, 
y 

5  microfarads  =  5  X  9  X  105  Elst  units  of  capacity, 

or  5  Elst  units  of  capacity  =  Q          5  microfarads. 

y  x  lu 

(7)  Approximation  Formula.*     Certain  other  Expressions. 

In  a  mathematical  expression  it  sometimes  occurs  that  some 
quantities  are  very  small  compared  with  others.  In  such  cases 
the  expression  may  often  be  given  a  form  which  is  more  convenient 
for  calculation  if  formulae  of  approximation  are  used. 

In  the  expressions  considered  let  a,  b,  c,  d,  etc.,  be  magnitudes 
which  are  very  small  as  compared  with  unity.  The  formulae  of 
approximation  may  then  be  given  forms  such  that  the  corrections 
are  contained  in  terms  which  are  added  to  or  subtracted  from  1. 

The  following  formulae  can  be  shown  to  hold  upon  the  above 
supposition,  and  will  often  be  found  convenient  to  use.  Where 
the  sign  =t  or  =F  is  used  before  a  quantity,  either  the  upper  or 
lower  sign  must  be  taken  all  thru  the  formula. 

In  general  (when  a  is  very  small  compared  with  1) 

(1  +  a)m  =  1  +  ma;  (1  —  a)m  =  1  —  ma.  (1) 

For  m  =  2 

(1  +  a)2  =  1  +  2a;  (1  -  a)2  =  1  -  2 a.  (2) 

For  m  =  \ 

Vl  +  a  =  1  +  }  a;  Vl  -  a  =  1  -  J  a.  (3) 

*  The  substance  of  what  is  given  under  this  head  has  been  selected  from 
"Physical  Measurements,"  by  Dr.  F.  Kohlrausch. 


MATHEMATICAL  QUANTITIES  AND  RELATIONS        373 

For  m  =  —  1 


For  m  =  -  2 

^-1-2-;  ^-1  +  2-.              (5) 
For  m  =  —  \ 

I                       -|             -i  J-                      ill                                      /CN 

.           =  1  —  |  a;  — T==  =  1  +  §  a.                (6) 

v  1  -f  a  v  1  —  a 

(1  ±  a)  (1  =b  6)(1  =b  c)  .  .  .    =  1  ±  a  d=  b  ±  c  .  .  .  (7) 


For  the  geometrical  mean  of  two  quantities,  which  are  very 
nearly  alike,  the  arithmetical  mean  may  be  used.     Thus 

.  (9) 


If  8  signifies  a  small  angle  measured  in  radians  (1  radian  =  57.2958 

degrees)  then, 

sin  (x  +  6)  =  sin  x  +  5  cos  x\  Sin  d  =  5,  (10) 

cos  (x  -\-  d)  =  cos  x  —  d  sin  x;  cos  5  =  1,  (11) 

tan  (x  +  d)  =  tan  x  +  -     -  ;     tan  d  =  d.  (12) 

COS    X 

Also,  if   a  quantity  a  is  very  small  compared  with  a  quantity 
x  >  1,  then 

loge  (x  +  a)  =  loge  x  +  -x  ;     loge  (1  +  a)  =  a.  (13) 

The  true  value  of  (a  +  6)n  is  given  by  the  expansion 

(a  +  6)"  =  an  +  na'-ty  +  "r^--  «n~2^>2  +  •  •  '        (14) 

The  exact  value  of  (1  d=  a)m  when  a2  <  1  is  given  by  the  ex- 
pansion 

m(m—  1)  m  (m—  l)(m  —  2)  a3 


h  •  •  •  (15) 


Any  quadratic  may  be  put  in  the  form  x2  +  px  =  q.     Its  solu- 
tion is  then 


374 


APPENDIX 


III. —  WIRE  DATA  AND  FORMULAE. 

(1)  Wire  Table  for  Pure  Copper;  from  Standard  Underground  Cable  Co. 

The  columns  of  this  table  are  reproduced  with  the  permission  of  the  Standard  Under- 
ground Cable  Co.,  of  Pittsburgh,  Pa.,  from  their  "  XVII  Hand-Book,  Standard  Underground 
Cable  Co.,  Copyright,  1906." 


B.  &S. 
G.  No. 

Diameter  in  mils 

;  Area  circular  mils 

Length,  feet  per 
ohm 

Resistance  in  Inter- 
national ohms  at 
68°F.  =  20°C. 
Ohms  per  1000  feet 

0000 

460.0 

211,600 

20,440 

0.04893 

000 

409.6 

167,800 

16,210 

0.06170 

00 

364.8 

133,100 

12,850 

0.07780 

0 

324.9 

105,500 

10,190 

0.09811 

1 

289.3 

83,690 

8,083 

0.1237 

2 

257.6 

66,370 

6,410 

0.1560 

3 

229.4 

52,630 

5,084 

0.1967 

4 

204.3 

41,740 

4,031 

0.2480 

5 

181.9 

33,100 

3,197 

0.3128 

6 

162.0 

26,250 

2,535 

0.3944 

7 

144.3 

20,820 

2,011 

0.4973 

8 

128.5 

16,510 

1,595 

0.6271 

9 

114.4 

13,090 

1,265 

0.7908 

10 

101.9 

10,380 

1,003 

0.9972 

11 

90.74 

8,234 

795.3 

1.257 

12 

80.81 

6,530 

630.7 

1.586 

13 

71.96 

5,178 

500.1 

1.999 

14 

64.08 

4,107 

396.6 

2.521 

15 

57.07 

3,257 

314.5 

3.179 

16 

50.82 

2,583 

249.4 

4.009 

17 

45.26 

2,048 

197.8 

5.055 

18 

40.30 

1,624 

156.9 

6.374 

19 

35.89 

1,288 

124.4 

8.038 

20 

31.96 

1,022 

98.66 

10.14 

21 

28.46 

810.1 

78.24 

12.78 

22 

25.35 

642.4 

62.05 

16.12 

23 

22.57 

509.5 

49.21 

20.32 

24 

20.10 

404.0 

39.02 

25.63 

25 

17.90 

320.4 

30.95 

32.31 

26 

15.94 

254.1 

24.54 

40.75 

27 

14.20 

201.5 

19.46 

51.38 

28 

12.64 

159.8 

15.43 

64.79 

29 

11.26 

126.7 

12.24 

81.70 

30 

10.03 

100.5 

9.707 

103.0 

31 

8.928 

79.70 

7.698 

129.9 

32 

7.950 

63.21 

6.105 

163.8 

33 

7.080 

50.13 

4.841 

206.6 

34 

6.305 

39.75 

3.839 

260.5 

35 

5.615 

31.52 

3.045 

328.4 

36 

5.000 

25.00 

2.414 

414.2 

37 

4.453 

19.83 

1.915 

522.2 

38 

3.965 

15.72 

1.519 

658.5 

39 

3.531 

12.47 

1.204 

830.4 

40 

3.145 

9.888 

0.9550 

1047.0 

No.  10  pure  copper  wire  weighs  31.43  pounds  per  1000  feet. 


WIRE  DATA  AND  FORMULAE  375 

(2)  The  Ohm. 

1  international  ohm  =  1.06300  Siemens  units. 

1  international  ohm  =  1.01348  British  Association  (B.A.)  units. 

1  international  ohm  =  1.00283  legal  ohms. 

1  international  ohm  =  resistance  at  0°  C.  of  column  of  pure 

mercury,  106.3  cms  long,  weighing 
14.4521  grams,  and  of  uniform  cross- 
section. 

1  British  Association  unit  (B.A.U.)  =  0.986699  international 

ohm. 

1  legal  ohm  =  0.997178  international  ohm. 

(3)  Resistance  of  Wire  Wound  in  a  Channel. 

The  following  formula  will  enable  the  resistance  to  be  calculated 
for  any  wire  with  any  thickness  of  insulation  when  wound  to  fill 
a  channel  of  any  given  volume.  The  formula  assumes  the  wire 
is  wound  in  square  order  and  in  a  regular  manner.  The  formula 
is  approximate  to  the  extent  that  terms  which  involve  the  square 
of  the  thickness  of  the  insulation  are  neglected  We  have 

1.27  Vp, 

R> ' 2F' 


In  relation  (1)  Rt  =  ohms  at  t°  to  which  wire  winds, 

V  =  total  volume  of  channel  filled  with  wire, 
Pt  =  resistivity  of  wire  at  t°, 
d  =  diameter  of  wire, 
h  =  twice  the  thickness  of  the  insulation. 

Example.  —  To  what  resistance  will  No.  26  B.  &  S.  copper  wire 
wind  per  cubic  centimeter,  when  its  temperature  is  20°  C.,  and 
the  double  thickness  of  its  insulation  is  5  mils? 

In  this  case 

V  =  1  cu.  cm, 
P20  =  1.594  X  1.08  X  10-*, 
d  =  0.0405  cm, 
h  =  0.0127  cm  (5  mils). 

These  values  placed  in  relation  (1)  give 
Rt  =  0.499  ohm. 


376  APPENDIX 

(4)  Certain  Formulae  for  Wire.* 

The  wire  is  pure  copper  and  at  20°  C.;  bare. 

(Ohms  per  1000  feet)  =  10354  -^  (area  in  cir.  mils). 

(Pounds  per  1000  feet)  =  0.0030269  X  (area  in  cir.  mils). 

(Feet  per  pound)  =  330,360  -r-  (area  in  cir.  mils). 

(Ohms  per  pound)  =  3,420,400  -f-  (area  in  cir.  mils)2. 

(Feet  per  ohm)  =  0.096585  X  (area  in  cir.  mils). 

2 

Area  in  circular  mils  =  diameter  in  mils  • 

For  approximate  calculations  it  may  be  easily  remembered  that 

1000  feet  No.  10  wire  =  1  ohm, 

1000  feet  No.  13  wire  =  2  ohms, 

1000  feet  No.  16  wire  =  4  ohms,  etc., 

1000  feet  No.    7  wire  =  \  ohm, 

1000  feet  No.    4  wire  =  \  ohm, 

1000  feet  No.    1  wire  =  J  ohm,  etc. 

To  change  microhms  per  centimeter-cube  to  ohms  per  mil-foot 
multiply  by  6.014.  To  change  ohms  per  mil-foot  to  microhms 
per  centimeter  cube  multiply  by  0.166.  (Statement  taken  from 
Catalogue  J,  Driver-Harris  Wire  Company.)  f 

IV.  — PHYSICAL  DATA. 

(1)  Resistivity  of  Mercury. 

The  following  data  was  furnished  the  author  by  the  Bureau  of 
Standards.  (In  his  own  work  the  author  has  used  the  value, 
volume  resistivity  of  mercury  at  20°  C.  =  95.782.) 

The  values  for  the  resistivity  of  mercury  are  those  computed 
from  the  temperature  formulae  of  F.  E.  Smith  of  the  National 
Physical  Laboratory  of  England  (Phil.  Trans.  204,  p.  112;  1904), 
C.  E.  Guillaume  of  the  International  Bureau  (Comptes  Rendus, 
115,  p.  414;  1892),  and  Kreichgauer  and  Jaeger  of  the  Reichsan- 
stalt  (Wied.  Ann.,  47,  527;  1892).  In  the  following  table  volume 
resistivity  is  given  in  microhms  per  centimeter  cube  and  mass 
resistivity  is  given  in  ohms  per  meter-gram. 

*  The  constants  under  this  heading  are  reproduced  with  the  permission 
of  the  Standard  Underground  Cable  Company,  of  Pittsburgh,  Pa.,  from  their 
"XVII  Hand-Book,  Standard  Underground  Cable  Company,  Copyright,  1906." 

t  Since  the  material  given  in  Appendix  III  was  prepared,  the  Bureau  of 
Standards  has  issued  circular  No.  31,  entitled  "  Copper  Wire  Tables."  This 
circular  should  be  consulted  for  the  most  authoritative  information  upon  wire 
sizes,  standards,  and  temperature  coefficients. 


PHYSICAL   DATA 


377 


Volume  resistivity 

Mass  resistivity 

0°C. 

20°  C. 

100°  C. 

0°C. 

20°  C. 

100°  C. 

Smith 

94.073 
94.073 

94.073 

95.783 
95.782 

95.782 

103.410 
103.379 

103.56 

12.7898 
12.7898 

12.7898 

12.9751 
12.9749 

12.9748 

13.8076 
13.8034 

13.828 

Guillaume  
Kreichgauer    and 
Jaeger  

The  0°  C.  values  are  computed  from  the  quantities  given  in  the 
definition  of  the  international  ohm,  14.4521  grams  mass  and 
106.300  centimeters  length;  and,  in  the  case  of  volume  resistivity 
at  0°  C.,  the  density  is  assumed  to  be  such  as  to  make  the  cross- 
section  1  mm.2  Smith's  temperature  formula  was  obtained  from 
observations  between  0°  C.  and  24°  C.,  Guillaume's  between  0°  C. 
and  61°  C.,  and  Kreichgauer  and  Jaeger's  between  14°  C.  and 
28°  C.  Accordingly  the  values  given  for  100°  C.  are  based  on 
extrapolation. 

The  temperature  formula  for  mass  resistivity  is  of  course  not 
the  same  as  for  volume  resistivity.  Thus,  letting  d  represent  mass 
resistivity,  p  volume  resistivity,  and  d  density, 

^  =  EL  .  $L. 
do       po     do 

The  temperature  formula  of  resistance,  as  measured  in  a  glass 
tube,  is  different  from  either  of  the  temperature  formulas  .for 
resistivity.  Thus,  letting  7  denote  the  resistance  as  measured  in 
a  glass  tube,  and  yg  the  linear  coefficient  of  expansion  of  the  glass, 

7t  =  Pt  .         1 

7o      po     1  +  ygt 

(2)  Resistivities  at  20 °C.;  Densities  and  Melting  Points  of  the 
Solid  Elements. 

Resistivity  or  the  specific  resistance  is  given  in  microhms.  (To 
change  to  ohms  multiply  by  10"6.)  The  resistivity  is  the  ohmic 
resistance  between  opposite  faces  of  a  centimeter  cube  of  the  sub- 
stance. Resistivity  is  quite  dependent  upon  the  purity  of  the 
material,  which  is  not  generally  recorded.  It  is  not  useful,  there- 
fore, to  record  values  beyond  three,  or  at  the  most  four,  figures. 
Where  values  given  by  different  observers  differ  a  mean  value 
is  given  here.  The  table  is  arranged  in  the  order  of  decreasing 
resistivity. 


378 


APPENDIX 


Element 

Resistivity 
in  mi- 
crohms at 
20°  C. 

Density, 
grams  per 
cubic  cen- 
timeter 

Melting  temperature, 
degrees  C. 

Bismuth  

119* 

9.80 

269 

Mercury  

95.  782  f 

13.5462 

-38.80 

Silicon  

59.5 

2.3 

1200  (?) 

Antimony 

41  3 

6  62 

629  2±0  5  F  P 

Thorium 

40  9± 

11  3 

1690 

Arsenic 

38 

5  73 

volatilizes 

Strontium  .  .  . 

25 

2  54 

900 

Tellurium  

21 

6  25 

450 

Lead  .  .                 

20  96 

11  37 

327 

Steel  (1%  C.)  

20 

7.8(?) 

Thallium  

19  § 

11.9 

301 

Tantalum  

14.7 

16.6 

2910(?) 

Rubidium  

12.3 

1.53 

38.5 

Iron  (0  1%  C.)    

12  1 

7  86 

1505  (about) 

Tin 

11  4 

7  29 

232 

Tungsten  (wire)  ... 

6-12 

18  8 

3002 

Palladium  .  .        

10  78 

11  4 

1549  2±2°  F.  P. 

Calcium  

10  5 

1  55 

780 

Platinum  

10  2 

21  5 

1755  0±5°  F.  P. 

Cobalt  

9.71 

8  6 

1489  8±2  0  F.  P. 

Nickel  

9.52 

8.9 

1452  3  ±2.  OF.  P. 

Osmium 

9  5 

22  5 

2200 

Lithium 

9  0711 

0  534 

186 

Indium 

9  0 

7  12 

155 

Cadmium                 .... 

7  57 

8  64 

320  2±0  3  F.  P. 

Potassium  .         

7  11U 

0  862 

62  5 

Zinc      

6  1 

7  1 

418  2±0.3F.P. 

Rhodium  

6  04 

12  44 

1907 

Iridium  

5  34 

22  41 

2290 

Sodium  

4  87H 

0  971 

97 

Magnesium  

4.57K 

1.723H 

633 

Molybdenum 

4  0 

8  6 

very  high 

Aluminum 

3 

2  65 

657 

Gold 

2  44 

19  32 

1062  4±0  8  F.  P 

Copper   . 

1  721 

8  89 

1082  6±0  8F.  P. 

Silver  (99.9%)  

1  65 

10  5 

960  0±0  7  F.  P. 

Barium  

3  75 

850 

Beryllium  

1  93 

1430 

Boron  

2.5(?) 

above  2000. 

Caesium  

1.87 

26.4 

Graphite,  carbon 

2  3 

vaporizes  at 

Lanthanum 

6  12 

about  3720. 
810 

Manganese 

7  39 

1200  (about) 

Neodymium 

6  96 

840 

Sulphur  (amorphous)*. 

insulator 

1  92 

444  6  (boils). 

Titanium  

3  54 

2500 

Vanadium  

5.5 

1620 

Zirconium 

4  15 

1300 

*  Changes  in  a  magnetic  field. 

t  See  Appendix,  IV,  1. 

J  Estimated.    Value  at  15°  C. 


40.1. 


§  Estimated.  Value  at  0°C.  =  17.6. 
||  Estimated.  Value  at  0°  C.  =  8.4. 
IT  Author's  value. 


PHYSICAL  DATA 


379 


The  density  of  a  substance  may  depend  upon  its  previous  treat- 
ment and  its  physical  state;  hence  values  are  generally  not  given 
closer  than  three  figures.  The  densities  given  are  the  grams  per 
cubic  centimeter  of  the  substance  at  room  temperature  and  at 
atmospheric  pressure.  When  authorities  differ  a  mean  value  is 
given. 

The  melting  point  is  given  in  degrees  centigrade  at  which  the 
substance  melts.  In  the  case  of  several  substances  the  melting 
points  are  very  sharp  and  have  been  determined  with  great  care. 
They  serve  as  fixed  points  of  temperature  which  are  used  for 
reproducing  the  temperature  scale.  In  these  cases  the  symbol, 
F.P.,  follows  the  recorded  temperature;  the  numerical  figure,  as 
db  0.8  states  that  the  recorded  temperature  of  melting  is  known 
within  that  number  of  degrees  centigrade. 

(3)  Data  on  a  Few  Alloys. 


Alloy 

Composition 

Maker 

Resis- 
tivity in 
mi- 
crohms 
at20°C. 

Temperature 
coefficient 
over  small 
range 

Density 

Driver-  Harris 

109  6 

0  00016 

8  02 

Nichrome  

Nickel  —  steel 

Wire  Co. 

96.6 
87  1 

0.00044 
low 

8.15 

8  14 

Constantan  
German  silver,  30% 

60Cu+40Ni 
30  Ni+Cu+Zn 

49 

48 

0.00001 
0  00023 

8.88 

Therlo  
Manganin  
German  silver,  18%  .  . 
Yankee  silver  
Platinoid 

Cu+Mn+Al 
84  Cu+4  Ni+12  Mn 
18  Ni+Cu+Zn 

62Cu+i5Ni+22Zn 

;;  

46.7 
44.5 
36 
33 
32  5 

0.0000056 
0.00002 
0.0003 
0.00028 
0  00025 

8.15 
8.5 
8.5 
8.6 
9 

Ferronickel  

« 

28.2 

0.002 

8.2 

Brass  

70  Cu+30  Zn 

6  to  9 

0.001 

8.  4  to  8.  7 

(4)  Standard  Solutions  for  Calibrating  Purposes.     (See  §  1121.) 
The  table  gives  the  resistivity  pt  at  six  temperatures  t  of  NaCl  t 

and  KC1  solutions. 

The  numbers -in  the  columns  are  resistivities.     They  express  the 

resistance  in  ohms  of  the  solution  (at  the  temperature  in  degrees  C. 

heading  the  column)  between  opposite  faces  of  a  centimeter-cube 

of  the  solution.     The  reciprocals  of  the  resistivities  expressed  in 

ohms  are  conductivities  expressed  in  mho,  cubic  centimeter,  units. 

[Note  that  the  resistivity  of  a  saturated  solution  is  roughly  one 

million  times  the  resistivity  of  a  pure  metal.] 

n  =  normal  solution. 


380 


APPENDIX 


Thus,  1  n  =  normal  KC1  =  74.59  gms.  of  the  salt  dissolved  in 
1  liter,  at  18°  C.,  of  pure  water. 

NaCl,  Sat.  =  saturated  solution  of  NaCl  at  temperature  head- 
ing a  column. 


Solution 

o°c. 

8°C. 

12°  C. 

16°  C. 

20°  C. 

24°  C. 

NaCl   Sat 

7  435 

5  924 

5  342 

4  847 

4  425 

4  062 

KC1,  In... 

15  ..288 

12.572 

11.509 

10.592 

9.797 

9.104 

KC1,  O.ln  
KC1,  0.02n  

139.86 
657.9 

112.61 
526.3 

102.14 

478.5 

93.283 
436.7 

85.690 
400.0 

79.113 
369.0 

KC1,  0.01  n  

1282.0 

1031.0 

934.6 

852.5 

782.47 

721.50 

(5)  Resistivities  of  Insulators. 

These  vary  so  widely  with  the  temperature,  the  purity  and 
character  of  the  material,  the  state  of  the  surface,*  and  the  con- 
ditions of  the  test,  that  the  values  given  below  are  only  roughly 
approximate  and  merely  serve  to  indicate  the  order  of  magnitude 
of  the  quantities.  Expressed  in  megohms  resistance  between 
opposite  faces  of  a  cubic  centimeter  of  the  material  the  resistivities 
of  a  few  materials  at  ordinary  temperatures  are: 

Paraffin  wax          =  3  X  1012, 

Mica  =  9  X  109, 

Ebonite  =  2  X  109, 

Porcelain,  50°  C.,  =  2  X  109, 

Rubber  (used  as  insulation  on  wire)  =  4.5  X  108, 
f  Fused  Silica  =  2  X  108, 

t  Glass  (soda-lime)  =  5  X  105. 


Consult  par.  815. 

National  Physical  Laboratory,  England. 


INDEX. 

(Numbers  refer  to  pages.) 


A. 

i  values  of,  363. 


1000 -a 

Accuracy,    checked    by    using    dif- 
ferent methods,  6. 
Ageing  of  thermometers,  "304. 
Alloys,  data  on,  379. 
"Alternating  Currents,"  Bedell  and 

Crehore,  205. 

Alternating    current,    resistance    de- 
termined with,  199. 
Alternating-current  resistance,  appa- 
ratus to  measure,  200. 
error  in  measuring,  considered,  206. 
remarks  upon,  199. 
test  of  method  of  measuring,  207. 
Aperiodic  motion,  348. 
Approximation  formulae,  372. 
Armagnat,  H.,  347. 
Arrangements  of  rheostat  coils,  81. 

five  plans  described  for,  81. 
Assumptions,  when  using  deflection 

instruments,  20. 
Ayrton  shunt,  160. 

errors  in  using,  considered,  164. 

full  theory  of,  160. 

galvanometer    constant    obtained 

with,  166. 
insulation  measurement  using  an, 

167. 
numerical  illustrations  in  using  an, 

165. 
table  relating  to,  161. 


B. 

Babcock,  H.  D.,  327. 
Balance,   systematic  method  of  ob- 
taining, 97. 
Batteries,  curves  for,  217. 


Batteries,  —  Continued. 
internal  resistance  of,  214. 
tested  by  condenser  method,  215. 
Battery,   resistance  of,    by   Kelvin's 

method,  235. 
resistance  of,  by  Mance's  method, 

218. 
resistance  of,  by  Siemen's  method, 

236. 
resistance    of,    by   voltmeter    and 

ammeter  methods,  220. 
Battery    resistance,    alternating-cur- 
rent   methods    of    measuring, 
226. 

bridge  method  of  measuring,  elec- 
trodynamometer  detector,  230. 
bridge  method  of  measuring,  tele- 
phone detector,  226. 
diminished  deflection  method,  233. 
galvanometer    deflection    methods 

of  measuring,  233. 
test  of  A.  C.  method  of  measur- 
ing, 228. 

Bedell  and  Crehore,  347. 
Bell  Telephone  Company,  288. 
Bell    Telephone    Company    method, 

248. 

Bell  Telephone  Company  test,  288. 
Bimetallic   conductors,    conductivity 

of,  368. 
Bridge,  Bureau's  form  for  comparing 

standards,  328,  329. 
Bridge  wire,  calibration  of,  70. 
Bureau  of  Standards,  4,  148,  299,  301, 

326. 

Burgess,  C.  F.,  29. 
Burgess,  method  of,  29. 
Bus-bar,  measurement  of  current  in, 

108. 
resistance  of,  106. 


381 


382 


INDEX 


C. 

Cable,  insulation  resistance  per  mile, 

169. 
Cables,  faults  on  heavy  underground, 

284. 
formula  for  calculating  insulation 

resistance  of,  185. 
insulation  resistance  of,  185. 
Calibration  of  bridge  wire,  70. 
application  of,  73. 
coils  used  for,  71. 
curve  for,  73. 
sample  measurement,  75. 
theory  of  method  of,  72. 
Callender  and  Griffiths,  298. 
Capacities,  comparison  of,  285. 
Capacity  and  resistance,  theorem  on 

relation  between,  186. 
Capacity  of  conductors  in  cables,  285. 
Cardew,  Major,  156. 
Cardew's  electrometer  method,  156. 
Carey-Foster  bridge,  mechanical  de- 
sign of,  65. 
standardization  of  bridge  wire  of, 

68. 
Carey-Foster    method,     connections 

for,  61. 
for    comparing    resistances    which 

differ  considerably,  66. 
methods    for    determining    bridge 

wire  resistance  in,  63. 
shunt  for  bridge  wire  in,  65. 
specimen  readings  made  with,  67. 
theory  of,  62. 

three  bridge  wires  used  in,  65. 
unique  feature  of,  61. 
uses  of,  65. 

Cell,  standard  cadmium,  3. 
Celluloid,  specific  resistance  of,  179. 
Celluloid    condenser,    resistance    of, 

176. 
C.G.S.  System,   conversion  of  units 

in,  371. 
Closed   circuit,   resistance  measured 

without  opening,  103. 
resistance  of  section  of,  102. 
Columbia  dry  cell,  resistance  tested, 
229 


Combinations  of  resistance  coils,  80. 

formulae  for,  80. 

Committee  on  dry  cell  test,  218. 
Condenser,  method  of  connecting  a 

variable,  182. 

resistance  of  celluloid,  176. 
standard,  3. 

Condenser   method   of   testing   bat- 
teries, 215. 
Condensers,    theory    of    leakage    of, 

170. 

Conductance,  defined,  132. 
Conductivity,  calculated  from  resist- 
ance data,  148. 
of  bimetallic  conductors,  368. 
commercial  measurement  of,  140. 
equipment  for  determining,  145. 
determined    with    variable   ratios, 

.  150. 
relation  to  temperature  coefficient, 

148. 

standards  of,  132. 

Conductivity    and    resistance,    rela- 
tions between,  367. 
Conductor  considered  as  a  conveyor 

of  energy,  133. 
Conductors,     general     and     specific 

properties  of,  136. 
money  value  of,  370,  371. 
Conjugate  conductors,  51. 
Connection,  methods  of  ascertaining, 

and  fault  location,  292. 
Constant,     arbitrary    galvanometer, 

168. 
Contact    resistances,    expression   for 

errors  due  to,  79. 
in  arms  of  Wheatstone  bridge,  78. 
Conversions  to  different  systems  of 

units,  371. 

"Cooling  curves,"  footnote,  345. 
Copper  wire,  table  for,  374. 

weight  of,  per  1000  feet,  374. 
Correction  factor,  errors  by  neglect- 
ing, 119. 
in  formula  of  Kelvin  double  bridge, 

118. 

Corrections,  methods  of  applying,  in 
loop  test,  278,  280. 


INDEX 


383 


Crosses,  method  of  locating  induc- 
tive, 288. 

Crosses  or  grounds,  loop  methods  for 
locating,  258. 

Curves,  of  galvanometer  motion,  348. 
of  insulation  resistance,  195. 

D. 

Damping  of  galvanometer  system, 
348. 

D'Arson,  a  unit,  356. 

Data  exhibited  by  curve,  5. 

Decade,  advantages  of  four  coils  to 

the,  83,  85. 

combinations  of  four  coils  to  give, 
84. 

Decade  plan,  advantages  of,  81. 

Deflection  indicator  for  measuring 
temperature,  318. 

Deflection  instruments,  low  accu- 
racy of,  16. 

Densities  of  solid  elements,  377. 

Density  of  copper,  134. 

Dial  bridges  for  temperature  meas- 
urements, 314. 

Differential  circuits,  application  of,  to 

thermometry,  46. 
practical  advantages  of,  45. 
proof  of  a  useful  property  of,  43. 
properties  of,  41. 

Differential  methods,  41. 
assumptions  made  in,  42. 

Differential  galvanometer,   for  tem- 
perature measurement,  312. 
used  with  shunts,  48. 

Diminished  deflection  method,  233. 

Dry  cells,  report  on  methods  of 
testing,  217. 

E. 

e,  functions  of,  365. 

Electric  wiring  system,  insulation  of, 
210. 

Electrical  measurement,  subjects  con- 
sidered under,  1. 

Electrical  Review,  reference  to,  79. 

Electrical  World  and  Engineer,  ref- 
erence to,  30. 


Electrodynamometer,  for  battery  re- 
sistance, 230. 
the  Rowland,  200. 
used  in  substitution  method,  231. 
Electrolytes,  cell  for  containing,  245. 
not  measurable  by  direct  current, 

238. 

resistance  of,  238. 
resistance  of,  by  method  of  Kohl- 

rausch,  240. 
resistivities  of,  244. 
special  bridge  for  measuring,  242. 
Electrostatic   induction,   relation  to 

other  quantities,  188. 
Energy,  money  value  of  electrical,  6. 
English-metric  conversions,  365. 
English  Post  Office  bridge,  89. 
Errors,  absolute,  8. 
accidental,  6. 

calculation  of,   applied  to  deflec- 
tion instruments,  14,  15. 
calculation  of,  extended  to  several 

•  variables,  10. 
gross,  5. 
mean,  8. 
per  cent,  9. 
relative,  8. 
systematic,  6,  7. 

Evaluation  by  reading  small  deflec- 
tions, 97. 


F. 

Factory  testing  set,  198. 
Fault,  distance  to,  in  loop  test,  260. 
faulty  wire  of  known  length,  two 
good  wires  of  unknown  length, 
267. 
located  with  faulty  wire  and  good 

wire  of  unknown  length,  277. 
Fault  finder,  a  lineman's,  294. 
Fault-locating  apparatus,  the  Fisher 

set,  293. 

Fault  location,  lead  wires  used  in,  281. 
practice  and  accuracy  in,  290. 
problems  in,  253. 

relation  of  principles  to  practice 
in,  253. 


384 


INDEX 


Fault  location,  —  Continued. 

two   faulty    wires    and    one   good 
wire  of  unknown  length,  276. 
Faults,  definition  of,  251. 
list  of,  252. 

location  of,  as  opens,  284. 
methods  of  locating,  251. 
upon  low-tension  power  cables,  283. 
Feussner,  decade  system  of,  87. 
Fiber,  specific  resistance  of,  183. 
Figure  of  merit,  defined,  354. 
of  galvanometers,  350. 
unit  of,  356. 

Fisher,  H.  W.,  267,  276,  293. 
Fisher's  loop   test,   modification   of, 

270. 

Fisher's    method,    loop    test    when 
length  of  cable  is  known,  269. 
special  case,  276. 
Five-coil  combinations,  86. 
Formulae  for  conductivity,  137,  138, 

139. 
Four-coil  arrangement  of  resistances, 

Northrup's  method,  82. 
Four-coil  arrangement  used  with  dial 

switches,  84. 

Franklin    Institute,    Jour,    of,    foot- 
note reference,  358. 
Fundamental  coefficient,  defined,  299. 
Fundamental  interval,  defined,  299. 

G. 

Galvanometer,  arbitrary  constant  of, 

168. 
constant  of,  obtained  with  Ayrton 

shunt,  166. 
current  reduction  thru,  by  Ayrton 

shunt,  162. 
description   of   a  high   sensibility, 

361. 

Einthoven  string,  357,  358,  359. 
flat-coil    pointer    type,    described, 

340. 

indicator  in  null  methods,  40. 
moving    magnet    and    D'Arsonval 

type  of,  339. 

portable  pointer,  used  with  ohm- 
meter,  58. 


Galvanometer,  —  Continued. 
portable     type,    for     temperature 

measurement,  311. 
proportionality  for  flat-coil  type, 

345. 
resistance  of,  measured  by  second 

property  of  bridge,  69. 
sensibility  of  pointer  type,  340. 
Galvanometer  constants,  table  of,  360. 
Galvanometer  shunts,  157. 
theory  of  ordinary,  157. 
Galvanometer  system,  "dead  parts" 

of,  357. 

equation  of  motion  of,  346. 
Galvanometers,  comparison  of,  349. 
diverse  qualities  of,  350. 
drawing  of  flat-coil  type,  341. 
flat-coil  type  of,  340. 
sensitive,  for  insulation  testing,  346. 
Gas,  expansion  of,  297. 
Graphite,  resistivity  of,  4. 
Ground,  located  upon  a  single  line, 

253. 

on  single  line,  sample  test,  257. 
testing  from  each  end  of  line  for, 

254. 

testing  from  one  end  of  line  for,  255. 
Grounds,  location  of,  on  high-tension 

cables,  281. 
resistance  of,  248. 

Grounds  or  crosses,  loop  methods  for 
locating,  258. 

H. 

Heraeus  platinum,  302. 
Herseus  wire,  purity  of,  307. 
Hering,  Dr.  Carl,  29. 
Bering's  conversion  tables,  7. 
Bering's  liquid  potentiometer,  247. 
High  resistance,  electrometer  method 

for  measuring,  175. 
leakage  method   I  for  measuring, 

171. 
leakage  method  II  for  measuring, 

175. 
leakage  method  III  for  measuring, 

180. 
measured  by  leakage  methods,  170. 


INDEX 


385 


High  resistance,  measured  by  method 

of  mixtures,  180. 
High    resistances,    capacity   used   in 

measuring,  155. 
deflection  methods  for  measuring, 

153. 

galvanometer  for  measuring,  157. 
measured  with  Wheatstone  bridge, 

153. 
metallic  resistances  excluded  from, 

152. 

specified  and  described,  152. 
two  general  classes  of,  152. 
without  appreciable  capacity,  157. 
High-tension     cables,      location     of 

grounds  on,  281. 
testing  outfit  for,  283. 
Hoopes  bridge,  140. 

operations  required  in  using,  143. 
precautions  in  using,  144. 
standards  for,  142. 
theory  of,  141. 

Humidity,    effect    of,   on    resistance 
standards,  326. 

I. 

Inductive  crosses,  location  of,  288. 
Inertia,  moment  of,  of  galvanometer 

system,  352. 
Insulation     measurements,     factory 

testing  set  for,  198. 
with    galvanometer    and    Ayrton 

shunt,  167. 
Insulation   resistance,    measured   by 

deflection  methods,  191. 
of  a  long  cable,  191. 
operations  required  in  measuring, 

by  deflection,  191. 
per  mile  of  cable,  169. 
Insulation  resistance  of  a  cable,  for- 
mula for  calculating,  185. 
Insulators,  resistivities  of,  380. 
Instruments,  distinction  between  in- 
dicator and  deflection,  338. 
proposed  treatment  of,  324. 

K. 

Kelvin  double  bridge,  a  network  of 
nine  conductors,  115. 


Kelvin  double  bridge,  —  Continued. 
assembled  from  resistance  boxes, 

126. 

correction  factor  in  formula  for,  118. 
diagrams  of,  116. 
formula  for,  118. 
formula  for  sensibility  of,  121. 
methods  of  applying  principle  of, 

123. 

Otto  Wolff's  ratio  coils  for,  123. 
ratio  coils  variable,  123. 
standard  variable,  125. 
theory  of,  117. 

Kelvin's  method,  for  measuring  gal- 
vanometer resistance,  69. 
Kelvin- Varley  slides,  75. 

equivalent  to  a  long  slide-wire,  77. 
Kempe's    Handbook    of    Electrical 

Testing,  53. 

Kohlrausch,  Dr.  F.,  30,  372. 
Kohlrausch's  method  for  resistivity 
of  electrolytes,  240. 

L. 

Landolt,  etc.,  246. 

LeChatelier,  306. 

Lead  wires,  resistance  of,  eliminated, 

310. 

used  in  fault  location,  281. 
Leakage  methods,  170. 

condition  for  highest  precision,  173. 
Leakage  of  condensers,  theory  of,  170. 
Leeds,  Morris  E.,  312. 
Leeds  and  Northrup  Co.,  58, 101,  119, 
126,  145,  146,  198,  200,  265, 
283,  340,  359. 

Lineman's  fault  finder,  294. 
Live    wires,    methods    of   measuring 

resistance  of,  210,  212. 
Loop  methods,  for  locating  grounds 

or  crosses,  258. 
H.  W.  Fisher's  method,  267. 
modified    to    meet    special    condi- 
tions, 267. 

the  Murray  loop,  258. 
Loop  test,  defined,  258. 

Fisher's    method    with    slide-wire 
bridge,  269. 


386 


INDEX 


Loop  test,  —  Continued. 

from  both  ends  of  faulty  wire,  271. 
illustrative  example,  274. 
modifications  of  Fisher's  method, 

269. 

position  of  galvanometer  in,  259. 
testing  from  one  end  when  return 
wire  is  of  unknown  resistance, 
273. 

testing   with   return   wire   of   un- 
known length,  271,  273. 
testing  from  one  end,  method  II,  274. 
Loop  tests,  correction  for  conductors 

of  different  sizes  in,  280. 
methods    of    applying    corrections 

in,  278,  280. 
Lord  Kelvin,  115. 
Low   resistance,    distinguished   from 

medium,  100. 
measured  with  ammeter  and  milli- 

voltmeter,  101. 

Low-resistance  standards,  set  of  pre- 
cision, 101. 

Low  resistances,  compared  by  Carey- 
Foster  bridge,  114. 
compared   by   modified   slide-wire 

bridge,  114. 

compared  by  potentiometer,  115. 
Low-tension   power   cables,   location 
of  faults  upon,  283. 

M. 

Magnesium,  conductivity  of,  139. 
resistivity  of,  127. 

Mains,  resistance  of  underground,  110. 

Mance's  method,  218. 

Manganin,     temperature    coefficient 
of,  101. 

Manganin  resistance,  5. 

Marine  cable  testing,  292. 

Marvin,  Prof.  C.  F.,  299. 

Matthiessen's  standards  of  conduc- 
tivity, 134. 
relation  between,  134. 

Maxwell's  rule,  96. 

Measurement,  plan  for  making  and 

recording  a,  126. 
sample  of  a  low  resistance,  127. 


"Measurement  of  Electrical  Resist- 
ance," Price,  155. 

Melting  points  of  solid  elements,  377. 

Mercury,  expansion  of,  297. 
resistivity  of,  376. 

Merit,  figure  of,  350,  354,  356. 

Metallic    conductors,    divided    into 
two  classes,  18. 

Methods    of    Chapter    II,    remarks 
upon,  37. 

Metric-English  conversions,  365. 

Mho,  defined,  133. 

Microvolt  D'Arson,  unit  defined,  359. 

Motion,  equation  of,  346. 

Multiple  arrangement  of  coils,  87. 

Murray  loop  test,  extension  of  slide- 
wire  for,  261. 

slide-wire  bridge  arrangement  for, 
260. 

N. 

National  Physical  Laboratory,  306. 
Network,   resistance   of,   when    con- 
taining E.M.F.'s,  224,  225. 
Wheatstone  bridge,  51. 
Nickel,  coefficient  of,  297,  299. 
use  of,  in  thermometers,  302. 
Nickel  resistance  change,  law  of,  300. 
Northrup's  four-coil  arrangement,  82. 
Null  methods,  6,  40. 

compared  with  deflection  methods, 
41. 

O. 
Ohm,  relation  to  other  units,  375. 

representation  of,  3. 

value  of  the,  375. 
Ohmic  resistance,  comments  on,  16. 

of  lamp  filaments,  17. 
Ohmmeter,  38,  58. 
Ohmmeters  and  meggers,  38. 

Mr.  Sidney  Evershed's  megger,  38. 

scales  of  meggers,  39. 
Opens,  located  when  good  wires  are 
available,  286. 

located  when  no  good  wire  is  avail- 
able, 287. 

location  of,  on  cables,  284. 


INDEX 


387 


Outfit,  conformity  in  the  parts  of 
an,  324. 

P. 

TT,  functions  of,  365. 

Paul,  of  London,  58,  340. 

Per  cent  Wheatstone  bridge,  93. 

Periodic  motion,  348. 

Photographic  record  of  temperature 
difference,  319. 

"Physical  and  Temperature  Con- 
stants," by  Kaye  and  Laby, 
246. 

Physical  data,  376. 

Platinum,  constancy  of,  301. 
deterioration  of,  306. 

Platinum  temperature,  defined,  298. 

Plugs,  correct  type  of,  336. 

Polarities,  a  word  on,  221. 
principle  of,  illustrated,  222. 

Porcelain  tube  for  thermometers, 
307. 

Position  for  galvanometer  in  Wheat- 
stone  bridge,  96. 

Potential  drops,  special  case  for 
comparing,  24. 

Potential  points,  used  with  low 
resistance,  100. 

Practice,  need  of,  in  measurement,  5. 

Precision,  degree  of,  4. 

Price,  W.  A.,  155. 

Pyrometry,  methods  of,  footnote 
reference,  323. 

Q. 

Quadratic  equation,  solution  of,  373. 
Quantities,  list  of  electrical,  1,2. 

R. 

Ratio  arms,  54. 

arrangements  of  resistances  for,  89. 
Schone's  arrangement  of,  91. 
simple  method  of  reversing,  89. 
without  contact  resistance,  92. 
Resistance  and  capacity,  when  prod- 
uct of,  is  constant,  189. 
Resistance    and    conductivity,    rela- 
tions between,  367. 


Resistance   boxes,    general   remarks, 
331. 

Resistance,  low,   introductory  state- 
ment to  measuring,  100. 

Resistance  of  closed  circuits,  general 
method,  102. 

Resistance    of    underground    mains, 

measured,  110. 
table  of  results,  113. 
test  of  method,  112. 

Resistance  measured,  with  voltmeter 

(power  on),  210. 
with  galvanometer  (power  on),  212. 

Resistance  measurements,  E.M.F.  in 
circuit,  210. 

Resistance  sets,  precision  of,  334. 

Resistance    spools,    construction    of, 
333. 

Resistance  standard,  Bureau's  form 

of,  327. 

constancy  of  Bureau's  form  of,  327. 
variable,    for    conductivity   deter- 
minations, 147. 

Resistance    standards,    variation   of, 
with  humidity,  326. 

Resistances  subdivided,  19. 

Resistance  units,   watt  capacity  of, 
332. 

Resistivities,  of  insulators,  380. 
of  solid  elements,  377. 

Resistivity,  defined,  132. 

Reversible  arms,  .when  useful,  90. 

Rheostat,  position  of,  in  bridge,  54. 

Rosa,  E.  B.,  326,  327. 

Rowland  electrodynamometer,  200. 

Royal  Berlin  Porcelain  Works,  307. 

Rymer-Jones  key,  use  of,  174. 

S. 

Sack,  Dr.  H.,  360. 
Schone's  arrangement  of  ratio  arms, 

91. 
Sensibility,  of  galvanometer  used  with 

Ayrton  shunt,  162. 
precisely  defined,  355. 
relation  of,  to  size  of  instrument, 

6. 
useful,  defined,  353. 


388 


INDEX 


Sensibility  and  accuracy,  discussed, 

325. 

Shunt,  Ayrton,  160. 
full  theory  of,  160. 
multiplying  power  of,  denned,  158. 
universal,  160. 
Shunt  boxes,  159. 

Shunts    used    with    differential    gal- 
vanometer, 48. 
Siemens  and  Halske,  360. 
Siemens'  method,  236. 
Silver-chloride  cells,  curves  for,  218. 
Slide-wire    bridge,     compared     with 

Kelvin  double  bridge,  60. 
for  reading  temperature,  308. 
for  resistance  of  electrolytes,  242. 
methods  of  using,  54. 
modified  for  comparing  low  resist- 
ances, 58. 

use  of  n  coils  with,  55. 
used     with     electrodynamometer, 

244. 

used  with  telephone,  243. 
with  long  wire,  242. 
Slide-wire    resistance,    arranged    for 

direct  reading,  58. 
circular  form  of,  58. 
method  of  making  a  high,  56. 
Smith,  Irving,  decade  system  of,  87. 
Sodium-chloride  solution,   resistivity 

of,  246. 
Solutions,    standard,  for   calibrating 

purposes,  379. 
Standard,  the  metergram,  134. 

the  metermillimeter,  134. 
Standard  low  resistances,  list  of,  124. 
Standard    resistance,     Reichsanstalt 

form  of,  329. 

the  Bureau's  form  of,  327. 
Standard  resistances,  capacity  of,  331. 

list  of  Otto  Wolff's,  330. 
Standard  solutions,  379. 
Standard  Wiring  Table  Committee, 

134,  136. 

Standards,  Bureau  of,  4. 
defined,  3. 
of  resistance,  326. 
precision  of,  4. 


Stewart,  O.  N.,  347. 

Substitution,  resistance  measured  by, 

27. 
Substitution  method  for  electrolytes, 

248. 

Sulphur,  boiling  point  of,  299. 
Survey  measurement,  5. 
Suspension,  tensile  strength  of,  344. 

T. 
Table,  illustrating  method  described 

in  par.  207,  34. 
Telephone,  the  differential,  50. 
Temperature,         direct        deflection 

methods  for  reading,  317. 
measurement    of    extremely    high, 

322. 

measured  with  dial  bridges,  314. 
measured     with      Kelvin     double 

bridge,  315. 

read  by  deflection  methods,  317. 
read  on  slide-wire  bridge,  308,  310: 
remarks  on,  296. 
Temperature     coefficients,    formulae 

for,  366. 
Temperature      difference,      accurate 

measurement  of,  319. 
curve  for,  322. 
diagram   of   measuring   outfit   for, 

321. 

photographic  record  of,  319. 
Temperature  measurement,  portable 

set  for,  311. 

Temperature  scales,  formulas  for,  366. 
Testing  set,  portable,  for  loop  tests, 

259. 
Theorem,  applied  to  measurement  by 

leakage,  189. 

on  relation  of  capacity  and  resist- 
ance, 186. 

Thermocouples,  308. 
Thermoelectric  force  against  copper, 

101. 

Thermometer  bridge  with  two  travel- 
ing contacts,  312. 

Thermometers,    constancy   of   resist- 
ance, 301. 
construction  of  resistance,  302. 


INDEX 


389 


Thermometers,  —  Continued. 
method  of  reading,  with  potential 

leads,  316. 
practical  methods   of   calibrating, 

300. 

protecting  case  for  resistance,  304. 
special  designs  of  resistance,  304, 

305,  306. 

upper  range  of  resistance,  306. 
uses  of  resistance,  303. 
with  potential  leads,  316. 
Thermometry,    electrical    resistance, 

297. 

remarks  on,  296. 
Time,  for  a  capacity  to  leak,  191. 
relation  of,  to  resistance  and  capac- 
ity, 173. 
Tube,  resistance  capacity  of  a,  241. 

resistance  of  a  conical,  241. 
Twist  of  wires  in  cables,  269. 


U. 

Underground  cables,  method  of  locat- 
ing grounds  upon,  284. 

Unit,  defined,  2,  3. 

Useful  sensibility  of  a  galvanom- 
eter, 353. 


V. 

Variable  resistance  standard,  145. 
Varley  formula,  modification  of,  267. 
Varley  loop,  described,  262. 
Varley  loop  test,  formulae  for,  263, 
264. 

illustrative  examples  of,  264. 

notes  on,  265. 

Velocity  of  light,  value  of,  371. 
Voltmeter,  method  I,  test  by,  20. 

method  II,  test  by,  23. 

method  of  using  shunt  with,  26. 

resistance  of  Weston  type  of,  338. 

tables  illustrating  test  by,  22,  24. 


Voltmeter,  —  Continued. 

unknown  resistance  in  circuit  with, 
22. 

Voltmeter  and  ammeter,    resistance 
measured  by,  35. 

Voltmeter  method,    with   three   un- 
known resistances  in  loop,  30. 

Voltmeter  methods,  limitations  of,  34. 

W. 

Waidner,  C.  W.,  323. 

Watt  capacity  of  coils  in  Wheatstone 

bridges,  97. 

Weighing,  a  null  method,  40. 
Weston,  Edward,  58,  340. 
Weston   voltmeter,    figure   of   merit 

of,  360. 
Wheatstone  bridge,  arranged  to  read 

in  per  cent,  93. 
first  property  of,  52. 
five-dial  type  of,  336. 
galvanometer  to  use  with,  95. 
manganin  coils  in,  99. 
network  of,  51. 

Otto  Wolff's  type  of,  illustrated,  337. 
possible  precision  of,  98. 
proof  of  first  and  second  proper- 
ties of,  52,  53. 
range  of,  considered,  98. 
remarks  upon  use  of,  94. 
second  property  of,  52. 
unbalanced,  53. 

Wheatstone     bridges,     general     re- 
marks, 331. 

Wire,  resistance  of,  wound  in  chan- 
nel, 375. 

Wire  data  and  formulae,  374. 
Wire  resistance,  approximate  calcu- 
lation of,  376. 

Wire  table  for  copper  wire,  374. 
Wolff,  Otto,  101,  151,  330,  337. 
Wright,  J.  W.,  265. 
Wunsch,  Felix,  his  method  of  locat- 
ing grounds,  284. 


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